Development of ultrafast camera-based single fluorescent-molecule imaging for cell biology

An ultrafast camera developed by Fujiwara et al. allows single fluorescent-molecule imaging every 33 µs with a localization precision of 34 nm (every 100 µs; 20 nm) and enables ultrafast PALM imaging of whole live cells.


Introduction
Extensive attention has recently focused on improving the spatial resolution of fluorescence microscopy, leading to the development of various types of super-resolution methods (Shcherbakova et al., 2014;Liu et al., 2015;Nicovich et al., 2017;Sahl et al., 2017;von Diezmann et al., 2017;Baddeley and Bewersdorf, 2018;Sigal et al., 2018;Sezgin et al., 2019;Lelek et al., 2021;Liu et al., 2022). In contrast, there have been limited efforts toward enhancing the temporal resolution of fluorescence microscopy, particularly in single fluorescent-molecule imaging (SFMI) and single-molecule localization microscopy (SMLM; Wieser et al., 2007;Jones et al., 2011;Huang et al., 2013;Hiramoto-Yamaki et al., 2014;Kinoshita et al., 2017). However, improving the temporal resolution is critically important (Balzarotti et al., 2017;Eilers et al., 2018;Koyama-Honda et al., 2020;Schmidt et al., 2021) for observing living cells, and particularly for investigating the dynamic movements and interactions of molecules at the single molecule level (Baboolal et al., 2016). This was demonstrated by our detection of hop diffusion of membrane molecules in the plasma membrane (PM) by enhancing the time resolution of single-particle tracking down to 25 µs (Fujiwara et al., , 2016Murase et al., 2004). However, this method necessitated the use of large (40-nm diameter) gold particles as probes. Since fluorescent probes are much smaller (<1 nm) and are used far more broadly than gold probes, in the present study, we aimed to improve the temporal resolution of imaging single fluorescent molecules (many molecules at the same time) to levels comparable to those of gold-particle tracking.
Here, we developed an ultrahigh-speed camera system that has enabled the fastest SFMI to date. We achieved a 100-µs resolution (10-kHz frame rate) with a 20-nm localization precision for single Cy3 molecules for a frame size of 14 × 14 µm 2 (256 × 256 pixels), and a 33-µs resolution (30-kHz frame rate) with a 34-nm localization precision for a frame size of 7.1 × 6.2 µm 2 (128 × 112 pixels). These frame rates are faster than normal video rate (30 Hz) by factors of 330 and 1,000, respectively (summarized in Table 1).
The ultrafast camera that we have developed simultaneously achieves high frame rates and reasonable photon sensitivity, allowing for a good balance between the temporal resolution and the localization precision. This balance is particularly important for high-speed imaging (for observing molecular dynamics in live cells), where observations have to be made near the photonlimited conditions due to fluorophore photophysics. Furthermore, we optimized the imaging conditions and fluorescent dye selection for high-speed simultaneous tracking of many single molecules for reasonable durations.
For tracking single molecules in living cells, we favor a camera-based method because it enables observations of several Table 1. Summary of the camera frame rate, maximal data acquisition image size (pixels), position determination precision, and number of observable frames for various fluorescent probes for the developed camera system and other cameras  Fujiwara et al. (2023) 0.033 512 × 512 (EM-CCD; STORM) Alexa647 l 7.2 i ND Dempsey et al. (2011) 0.8 2,048 × 256 (sCMOS; STORM) Alexa 647 m ≈10 ND Lin et al. (2015) 1.6 2,048 × 128 (sCMOS; STORM) Alexa 647 m ≈12 ND Lin et al. (2015) tens to hundreds of molecules simultaneously in a given field of view, like in a cell. This approach is particularly advantageous for investigating the interactions and assemblies of molecules in live cells and distinct behaviors of molecules in various regions of the cell. In addition, the highly sensitive ultrafast cameras can dramatically accelerate the data acquisitions for single-molecule localization microscopy methods, such as PALM and dSTORM, as described in the companion paper. We examined the applicability of this camera system to observations of rapid molecular movements in the PM of live cells, by testing whether it can detect the hop diffusions of Cy3labeled single molecules, a phospholipid and a transmembrane protein, transferrin receptor (TfR). Previously, the detection of hop diffusion of phospholipids and membrane proteins was only possible using 40-nm diameter gold probes and high-speed bright-field microscopy (single particle tracking), at a camera frame rate of 40 kHz (a time resolution of 25 µs Fujiwara et al., 2016;Murase et al., 2004] or 100 µs [Chai et al., 2022]). However, this could only be achieved in the apical (dorsal) PM, because the gold probes cannot enter the space between the basal (ventral) PM and the coverslip. We here report that ultrafast single-molecule imaging of fluorescently labeled molecules based on the developed ultrafast camera successfully reproduced the ultrafast gold-particle tracking data in the apical PM, while maintaining cell viability. Furthermore, when coupled with improved analysis methods developed here (Supplemental theories 1 and 2 in the Supplemental text), for the first time, we were able to directly evaluate the residency time distributions of membrane molecules within a compartment and perform a diffusion anomaly analysis over a 5 orders of magnitude time range (from 0.044 ms to 2 s). These novel findings provided further supports for the picket-fence model, based on the actinbased membrane skeleton meshes (fences) and transmembrane picket proteins anchored to and aligned along the fences Fujiwara et al., 2016;Morone et al., 2006).
With the development of ultrafast SFMI and tracking, we could experimentally address whether the molecular dynamics and compartmentalization in the basal PM are similar to those in the apical PM, using the fluorescently labeled phospholipid and transmembrane proteins. Our results demonstrate that both phospholipids and transmembrane proteins, TfR and EGF receptor (EGFR), undergo hop diffusion among the compartments in the bulk basal PM, with virtually the same compartment sizes (108 nm) and hop frequencies (once every 10 and 24 ms on average for the phospholipid and transferrin receptor, respectively) as those in the apical PM. These results indicate that, although the basal PM is located close to the coverslip, within 40 nm from the coverslip, and binds to it via focal adhesions in the basal PM, the basic structures and molecular dynamics in the bulk basal PM are very similar to those in the apical PM. Furthermore, in the companion paper (Fujiwara et al., 2023), we demonstrate that membrane molecules undergo hop diffusion even inside the focal adhesion, a micron-scale structure formed in/on the basal PM responsible for cellular attachment to and migration in/on the extracellular matrix.
Our new ultrafast camera system enables the data acquisition for PALM and dSTORM at accelerated rates of up to 1 kHz, which is ≈30× faster than standard rates, with 29 and 19 nm localization precisions, respectively, for large view-fields that can include an entire live cell. These developments will be described in the companion paper (Fujiwara et al., 2023).

Results
Basic design of the ultrafast camera system Our camera system is composed of a microchannel-plate image intensifier (Fig. 1, A a), a high-speed complementary metal-oxide semiconductor (CMOS) sensor (Fig. 1, A b), and an optical-fiber bundle, which couples the phosphor screen of the image intensifier to the CMOS sensor chip (Fig. 1, A c; Materials and methods). The image intensifier (Hamamatsu, V8070U-74) comprises a third-generation GaAsP photocathode with a quantum efficiency of 40% at 570 nm (Fig. 1, A a α), a threestage microchannel plate allowing maximal gain over 10 6 , with virtually undetectable non-linear noise increase (Fig. 1, A a β), and a P46 phosphor screen with a decay time of ∼0.3 µs from 90 to 10% (Fig. 1, A a γ). A CMOS sensor (the sensor developed for a Photron 1024PCI camera) with a global shutter exposure was used (Table 1).
A straight (1:1) optical-fiber bundle (Fig. 1, A c) was directly adhered to the phosphor screen of the image intensifier on one side (Fig. 1, A a γ, input side) and to the CMOS sensor on the other side (Fig. 1, A b, output side). This approach enhanced the signal reaching the CMOS sensor by a factor of 5-10, as compared with the lens coupling used for our standard camera system employed for SFMI Kinoshita et al., 2017). The electrons generated at the CMOS sensor were transferred, amplified (Fig. 1, A d), and digitized ( Fig. 1, A e) in 64 and 8 parallel paths, respectively, and subsequently transferred to the host PC.
Basic conceptual strategy to address the large readout noise of the CMOS sensor We opted to use a CMOS sensor instead of the more common scientific CMOS (sCMOS) sensor in fluorescence microscopy due to its higher frame rates, which can reach 10-100 kHz (10,000-100,000 frames per second), compared to sCMOS sensors. However, the readout noise of the CMOS sensor is generally much greater than that of the sCMOS sensor by factors of several 10s, making CMOS sensors rarely used in fluorescence microscopy. Therefore, to employ the CMOS sensor for SFMI, the problem of high levels of readout noise had to be solved.
Our basic conceptual strategy to address this problem was the following: we should place an amplifier before the noisy detector, so that both the input signal and noise generated at the photocathode can be amplified (they will be amplified in the same way by the amplifier) at least to the level that the amplified noise (be mindful that this must be the noise and not the signal) becomes comparable to the noise that the detector generates. For the CMOS camera (noisy detector), we placed an image intensifier (more specifically the microchannel plate; a-β in Fig. 1 A), before the CMOS sensor (b in Fig. 1 A), so that the noise (including the background) generated by the photocathode of the image intensifier (the first step of light detection; a-α in Fig. 1 A) is amplified at least to a level comparable to the readout noise of the CMOS sensor (signal at the photocathode is amplified by the same factor as the noise, but in the present argument, the amplification of the noise is the critical issue). Here, we explain why this basic strategy would work.
We denote the average intensities of the signal and noise + background at the photocathode of the image intensifier as S p and N p , respectively, that of the readout noise as Nr, and the image intensifier amplification gain as G. Note that when using the CMOS sensor for SFMI, generally N p << Nr. The signal-to-noise (including the background intensity) ratio (S/N) of the output from the CMOS sensor can be written as a function of G, (1) This is because both the signal and noise at the photocathode stage are amplified by a factor of G, neglecting the noise generated by the amplifier. Note that when there is no gain; i.e., G = 1, because N p << Nr. Namely, the S/N ratio of the CMOS sensor output is dominated by the large readout noise of the CMOS sensor Nr, and we will not be able to detect the single-molecule signal, S p , due to the large Nr (i.e., S p /N (G = 1) <1).
With an increase in gain G, Nr/G becomes much smaller than N p (or Nr << N p ×G; namely, when the noise at the photocathode is amplified so that the amplified noise becomes greater than the readout noise of the CMOS sensor Nr), and Eq. 1 can be written as where ε represents the very small value of Nr/G. Eq. 3 shows that with an increase of the amplifier gain G, the S/N of the output signal from the CMOS sensor is no longer dominated by Nr, and thus increased close to that at the photocathode (S p /N p ). Eq. 3 is consistent with the general truth that amplifier placement cannot increase the output S/N to more than the input S/N. Note that, in this argument, the key is not to compare S p with Nr, but to compare N p (or N p ×G) with Nr. Namely, when the detector itself is a large noise generator and the limiting factor for the S/N of the entire system, this problem can be minimized by placing an amplifier before the noisy detector. Eqs. 1 and 3 indicate that larger gains are preferable. However, since the amplifier itself can also generate noise and since the full-well capacity of each pixel of the CMOS sensor is limited (i.e., to make this strategy work, the dynamic range of the camera system must be quite large), there is an upper limit for the amplifier gain.
When we set the gain G so that N p ×G = Nr (the noise at the photocathode N p is amplified to become comparable to the readout noise Nr); i.e., if we set G = Nr/N p , Namely, under these conditions, the output S/N of the CMOS sensor is half the S/N at the photocathode output. This provides a rule of thumb that the noise at the photocathode (N p ) should be amplified at least to a level comparable to the readout noise of the CMOS sensor (Nr). For the actual values of Nr and N p , see the subsection "Ultrahigh-speed intensified CMOS camera system: Design and operation, Basic design concept of the camera system, (b) The use of an image intensifier" in Materials and methods.
Under these conditions, the sensitivity limitation for this camera system is determined by the quantum yield of the photosensor, which is ≈0.4 at ≈580 nm (for dyes including Cy3, Alexa555, JF549, and TMR). The typical photosensor quantum yield of the sCMOS cameras frequently used for SFMI is often ≈0.95, whereas the frame rate can go up to only ≈800 Hz for a frame size of 256 × 256 pixels (for example, Hamamatsu ORCA-Fusion BT). Therefore, in essence, we increased the speed by a factor of more than 12.5 (10 kHz) and up to 56.3 (45 kHz) by sacrificing the sensitivity by a factor of ≈2.4.
Apart from the basic concept, to determine the useful amplifier gain for experiments, we obtained the actual image readout noise, images of single photons acquired as a function of the intensifier gain, stochastic gain variations (fluctuations) of the image intensifier, and the probability of detecting single photons, as described in Fig. S1 and the subsection "Ultrahighspeed intensified CMOS camera system: Design and operation, Basic design concept of the camera system, (b) The use of an image intensifier" in Materials and methods. Based on our findings, we selected an overall electron amplification gain of 8,100×, which provides a 90.0% probability of detecting a photon-converted electron emitted at the photocathode of the image intensifier ( Fig. S1; Materials and methods).
Our developed camera system is typically operated at frame rates of 10 and 30 kHz (100-and 33-µs resolutions) with frame sizes of 256 × 256 and 128 × 112 pixels, respectively. However, the camera can be operated at faster rates with reduced image sizes. For example, it can be operated at 45 kHz or every 22 µs, with a frame size of 7 × 3.5 µm 2 . Virtually all of the single Cy3 molecules immobilized on coverslips that were observed at 60 Hz were also detectable at frame rates of 10 and 30 kHz ( Fig. 1, B-D).
Ultrafast imaging of many single molecules simultaneously at 10 and 30 kHz With the developed ultrafast camera system, the crucial issue for ultrafast SFMI is the number of detected photons emitted from a single fluorescent molecule during a single frame time. This is because the single-molecule localization precision is fundamentally determined by the number of detected photons/molecule during the camera's single frame time, as described by the Mortensen equation (Mortensen et al., 2010; refer to the legend for Fig. S2). We have thus assessed whether a sufficient number of photons can be emitted from single fluorescent molecules of various fluorophores during the 33 and 100 µs frame times of the developed camera system (30 and 10 kHz, respectively; neglecting a readout time of 812 ns). The plots of single-molecule localization precision vs. the number of detected photons/molecule during a single-frame time (using fluorescent molecules bound to the coverslip) were consistent with the Mortensen equation for all dye molecules examined here (Fig. S2). These plots demonstrated that the number of detected photons/molecule/frame and the single-molecule localization precision were enhanced with an increase of the excitation laser intensity, but to a limited extent (Fig. 1,E and F;and Fig. S3). This occurs because at higher excitation light intensities, the number of photons emitted from a single fluorescent molecule during a frame time is limited by the triplet bottleneck saturation (see the subsection "Estimation of the number of photons that can be emitted by a single Cy3 molecule during 0.1 ms: Triplet bottleneck saturation" in Materials and methods). The saturation became apparent from around 23 µW/µm 2 for Cy3 (Fig. 1,E and F;and Fig. S3). Among the eight dyes we tested, Cy3 exhibited the lowest tendency to saturate (Fig. S2 B and Fig. S3 A). Therefore, we primarily employed Cy3, but also utilized tetramethylrhodamine (TMR) as a membrane-permeable probe.
The best single-molecule localization precision achieved with the Cy3 fluorophore at a frame integration time of 0.1 ms (10-kHz frame rate) was 20 ± 0.71 nm, using total internal reflection (TIR) laser illumination at a density of 79 µW/µm 2 at the sample plane (n = 50 molecules; mean ± standard error of means [SEM] are given throughout this report; Fig. 1 F;and Fig. S2,A and B;and Fig. S3,B and C). At 30 kHz, since fewer photons were emitted from a Cy3 molecule during a frame time of 33 µs, the localization precision was worse (34 ± 1.0 nm; n = 50 molecules;  Table 1). The best single-molecule localization precision achieved with this camera system was 2.6 ± 0.099 nm, with a maximum number of detected photons/molecule/frame of 11,400 ± 700, at a frame integration time of 16.7 ms (60-Hz frame rate) in which most of the Cy3 molecules were photobleached (within a single frame; n = 50 molecules; Fig. 1,G and H;and Fig. S2,C and D;summarized in Table 1).
Taken together, we conclude that the observation frame rates of 10-30 kHz (with localization precisions of 20-34 nm, using the view-fields of 256 × 256-128 × 112 pixels [14.1 × 14.1-7.1 × 6.2 µm 2 ]), represent the ultimate fastest frame rates employable for single-molecule tracking in living cells, using the currently available fluorescent molecules. Meanwhile, the camera itself could be operated much faster. The Photron 1024PCI CMOS sensor, which was used most extensively here, can be operated at 10 kHz (256 × 256 pixels), 45 kHz (128 × 64 pixels; or 256 × 256 pixels using the newer SA1 sensor), and 110 kHz (128 × 16 pixels; summarized and compared with the results with other cameras in Table 1; Materials and methods, "Ultrahigh-speed intensified CMOS camera system: Design and operation"). Therefore, the camera is no longer the limitation for the faster frame rates, and with the future development of fluorescent dyes with higher throughputs, we should be able to perform single-molecule tracking at even better time resolutions. In fact, single transferrin (Tf) molecules bound by an average of 5.0 Cy3 molecules (5xCy3-Tf) adsorbed on the coverslip could be imaged and tracked at 45 kHz, with localization precisions of 30 (38) and 29 (34) nm, using TIR (oblique-angle) illuminations of 43 and 79 µW/µm 2 , respectively (  Table 1).

TIR and oblique illuminations for ultrafast SFMI
In this study, we employed both TIR and oblique-angle laser illuminations. TIR illumination is useful for observing single molecules in the basal (ventral, bottom) PM by suppressing the signals from the cytoplasm and the enhanced evanescent electric field for the same laser intensity. The TIR illumination provides better localization precisions than the oblique-angle illumination at the same laser intensities (27 vs. 38 nm; Fig. 1 F).
On the other hand, the oblique-angle illumination is more versatile: it can be used to illuminate the apical PM, endomembranes, and cytoplasm, which cannot be accomplished using the TIR illumination. Particularly, in this research, since we hoped to compare the present single fluorescent-molecule tracking results of the phospholipid diffusion with our previous 40-kHz single gold-probe tracking data obtained in the apical PM Fujiwara et al., 2016;Murase et al., 2004), we had to use the oblique-angle illumination. Therefore, as our "standard test conditions" in the present research (the end of the caption to Fig. S3), we employed the oblique-angle illumination with a laser intensity of 23 µW/µm 2 at the sample plane (the dye saturation starts around this laser intensity; Fig. 1 ,E and F;and Fig. S3), despite its worse localization precisions as compared with the TIR illumination.

Trajectory length and cell viability under the standard conditions
Under the standard test conditions (oblique-angle illumination at 23 µW/µm 2 ), single Cy3 molecules immobilized on the coverslip could be observed at 10 kHz with an average signal-tonoise ratio (SNR) of 2.5 ± 0.11 (Fig. 2,A and B;and Video 1), and the fractions of the trajectories with durations (uninterrupted length of the trajectories) longer than 100, 300, and 1,000 frames (after the 3-frame gap-closing, which neglects the nondetectable periods lasting for three frames or less) were 14, 2.9, and 0.31%, respectively, among all obtained trajectories (Fig. 2 C). Therefore, performing single-molecule tracking for 100-300 frames (10-30 ms at 10 kHz) is quite practical. Meanwhile, under the conditions for the best localization precision with TIR illumination of 79 µW/µm 2 (20 ± 0.71 nm), the fractions were 1.6, 0.13, and 0.0%, respectively ( Fig. 2 C), making it difficult to obtain trajectories longer than 100 frames (for the summary and comparison with results using other cameras, see Table 1).
These results indicate that, for successful single-molecule tracking, a proper compromise between the localization precision and the trajectory length is necessary. This is one of the major differences between single-molecule tracking and SMLM. SMLM basically requires only one localization for a single molecule (observations lasting only for a single frame) before photobleaching, while single-molecule tracking needs many localizations to acquire long trajectories (essentially, longer is better) before photobleaching/photoblinking.
Under the standard test conditions, the laser illumination (oblique-angle at 23 µW/µm 2 of the 532-nm line) for 1 min hardly affected the cell viability, although half of the T24 cells did not survive after 10-min irradiation (Fig. 2, D and E; human epithelial T24 cells were employed throughout this research and their microscope observations were always conducted at 37°C). Since all our ultrafast measurements were completed within 5 s, we conclude that the toxic effect of the illumination laser is minimal for our observations. In a previous report (Wäldchen et al., 2015), the light toxicity was found strongly dependent on the cell type, and our result using T24 cells is similar to their result using HeLa cells. However, the direct comparison of the present result and the previous result is difficult due to the differences in the viability test method, laser wavelength (532 vs. 514 nm), and illumination durations (1 and 10 vs. 4 min; our result suggests non-linear dependence of the viability on the illumination duration).
Testing the ultrafast camera system using PM molecules undergoing hop diffusion 1: Qualitative observations As stated in the Introduction, we previously detected non-Brownian diffusion (hop diffusion) of phospholipid and transmembrane protein molecules in the PM, using large (40-nm diameter) gold particles as a probe Fujiwara et al., 2016;Murase et al., 2004;Kusumi et al., 2005;Kusumi et al., 2012a;Kusumi et al., 2012b; also see Sheetz, 1983). This was made possible by improving the time resolution of single-particle imaging using bright-field microscopy down to 25 µs (Fujiwara et al., , 2016Murase et al., 2004). The hop diffusion is influenced by modulating the actin-based membraneskeleton meshes (Fig. 3 A). Therefore, we concluded that the entire apical PM is compartmentalized, by the steric hindrance of actin-based membrane-skeleton meshes (fences) and the steric hindrance + hydrodynamic friction-like effects from rows of transmembrane-protein pickets anchored to and aligned along the actin fence ( Fig. 3 A). In the compartmentalized PM, both transmembrane proteins and lipids undergo short-term The distributions of the durations of the on-periods and those after neglecting the offperiods (non-emission periods) lasting for 1, 2, or 3 frames (gap closing; see Materials and methods) of single Cy3 molecules immobilized on a coverslip and excited by oblique-angle illumination at 23 µW/µm 2 (standard conditions; left) or by TIR illumination at 79 µW/µm 2 (right), observed at 10 kHz (totals of 264 and 593 molecules, respectively). The three-frame gap closing was employed in single-molecule tracking under the standard test conditions (thick orange curve; left). (D and E) Photo-induced damage to the cells during ultrahigh-speed SFMI is very limited under our standard experimental conditions. Cell viability was examined by staining with 1 µM TOTO-3 iodide, which selectively stains dead cells, at 37°C for 5 min, and then observing the stained cells using epiillumination with a 594-nm laser. (D) Representative images of the nuclei stained with TOTO-3 iodide. Control, a reference image of a living cell (n = 24 images). H 2 O 2 , a reference image of a dead cell after the treatment with 100 µM H 2 O 2 at 37°C for 1 h (n = 48 images). (E) Histograms showing the fluorescence intensity of TOTO-3 in the 5.5 × 5.5-µm area inside the nucleus (see the square box in D; n = the number of examined cells). Top: Live cells without laser illumination (negative control). Second: Dead cells after the H 2 O 2 treatment (positive control). Based on the results of the negative and positive controls (top and second boxes, respectively), a threshold fluorescence intensity of 4.0 × 10 5 (arbitrary unit = AU) was selected to categorize the live and dead cells (96% [90%] of the negative [positive] control cells were categorized as alive [dead]). Third and Fourth: Cells were subjected to illumination under our typical 10-kHz single Cy3 molecule imaging conditions (oblique-angle 532-nm laser illumination at 23 µW/µm 2 ) for 1 and 10 min, respectively, which are longer by factors of 12 and 120 than our longest illumination duration of 5 s/cell (10 500-ms image sequences = 5 × 10 4 frames). We concluded that the light-induced damage to the cells is insignificant under our standard experimental conditions. confined diffusion within a compartment plus occasional hop movements to an adjacent compartment, which was termed hop diffusion Kusumi et al., 2012a;Jacobson et al., 2019).
Each gold-tagged molecule in the PM exhibited two distinct diffusion coefficients: the microscopic diffusion coefficient (D micro ), which characterizes the unhindered diffusion within a compartment, and the macroscopic diffusion coefficient (D MACRO ), which is largely determined by the compartment size and the hop frequency across intercompartmental boundaries composed of the picket-fence. D MACRO is significantly smaller than D micro .
The compartment sizes were in the range of 40-230 nm depending on the cell type (≈110 nm in human epithelial T24 cell line used here), and the dwell lifetimes of membrane molecules within a compartment are in the range of several to a few 10s of milliseconds (Murase et al., 2004;Fujiwara et al., 2016). Therefore, the hop diffusion of membrane molecules in the apical PM appeared to be quite suitable for testing SFMI using the developed ultrafast camera system. Indeed, this project was originally undertaken to develop a fast single fluorescent-molecule imaging method to observe the hop diffusion of membrane molecules in the PM. We sought to examine whether we could detect the hop diffusion of L-α-dioleoylphosphatidylethanolamine (DOPE) conjugated with Cy3, Cy3-DOPE, at a time resolution of 0.1 ms (10 kHz, 333 times faster than normal video rate) and transferrin receptor (TfR) tagged with Cy3-conjugated transferrin (average dye to protein molar ratio of 0.2) at a frame rate of 6 kHz (0.167 ms resolution; 200 times faster than normal video rate; slowed from 10 kHz because the hop rate of TfR was expected to be slower than that of Cy3-DOPE). The observations were made in the apical PM using the oblique-angle illumination (Fig. 3, B-D; and Videos 2, 3, and 4), for making the direct comparisons with the previous single gold-particle tracking data. All of the cell experiments reported here were performed at 37°C using human epithelial T24 cells (previously called ECV304 cells).
Virtually all of the single-molecule images and trajectories ( Fig. 3, B-D; and Videos 2, 3, and 4) gave the impression that the Cy3-DOPE and TfR molecules underwent rapid diffusion within a confined domain of ≈100 nm and occasionally moved out of this domain, became confined again at the place where it moved to, and repeated such behaviors. These typical behaviors appear to reproduce the movement of gold-tagged DOPE and TfR quite well.
Testing the ultrafast camera system using PM molecules undergoing hop diffusion 2: Quantitative and statistical analyses Apart from the subjective impression, the trajectories of single fluorescent molecules were examined quantitatively and statistically, using the plot of mean-square displacement (MSD) vs. time interval (Δt; Qian et al., 1991; called the MSD-Δt plot, which also provides single-molecule localization precisions; Fig. 4 A and Fig. S4). The results were compared with those obtained by single gold-particle imaging.
Based on the MSD-Δt plot (Fig. 4, A a), each trajectory could be classified into a suppressed-, simple-Brownian-, or directed-diffusion mode ; see the caption to Fig. 4, B a). More than three-quarters of the Cy3-DOPE and TfR trajectories obtained in the intact apical PM at 10 and 6 kHz (0.1 ms and 0.167 ms resolutions, respectively) were categorized into the suppressed-diffusion mode (top boxes in Fig. 4, B b; also see Table 2), reproducing the results obtained by single gold-particle tracking at 40 kHz (Table 2, in which all of the statistical parameters and the statistical test results for the diffusion parameters are also summarized).
Meanwhile, in the f-actin-depleted blebbed PM, almost all of the Cy3-DOPE and TfR trajectories, obtained at 10 and 6 kHz, respectively, were classified into the simple-Brownian-diffusion mode (Fig. 4, A c and B b; and Table 2), again consistent with the data obtained by single gold-particle tracking in the blebbed PMs of NRK cells (Fujiwara et al., , 2016. As concluded previously, these results indicate that the actin-based membrane skeleton is involved in inducing the suppressed diffusion of both phospholipids and transmembrane proteins Fujiwara et al., 2016;Murase et al., 2004;Hiramoto-Yamaki et al., 2014).
The MSD-Δt plot obtained for each trajectory was then fitted, using an equation based on the hop-diffusion theory (Kenkre et al., 2008;called "hop-diffusion fitting";Fig. 4 A; see "Supplemental theory 2. Hop-diffusion fitting: the function describing the MSD-Δt plot for particles undergoing hop diffusion" in the Supplemental text). The hop-diffusion fitting provided two diffusion coefficients for each trajectory observed in the PM: D micro and D MACRO . The presence of two diffusion coefficients for a single trajectory observed in the PM and only one diffusion coefficient in the actin-depleted blebbed PM (Fig. 4 C) again reproduced the observations obtained using single gold-particle imaging Fujiwara et al., 2016).
The hop-diffusion fitting of each trajectory obtained in the intact PM also provided the average compartment size L for each trajectory (Fig. 4 D and Table 2). Importantly, the Ls for Cy3-DOPE and TfR were quite similar, with median values of 101 and 103 nm, respectively (non-significant difference; in the following discussions and calculations, we will use a compartment size of 100 nm). This result suggests that the underlying mechanisms for confining phospholipids and transmembrane proteins are based on the same cellular structures, perhaps by the actinbased membrane-skeleton meshes (fences) and the transmembrane picket proteins bound to and aligned along the actin-fence (Fig. 3 A). The compartment sizes L found here (median sizes of 101 and 103 nm for Cy3-DOPE and TfR, respectively) are quite comparable to those previously detected using gold-conjugated molecules in the same T24 cells (110 and 100 nm for gold-tagged DOPE and TfR, respectively; Table 2).
Taken together, our ultrafast SFMI based on the ultrafast camera system developed here could successfully detect the hop diffusion of membrane molecules, which was previously detectable only using gold probes and high-speed bright-field microscopy. Therefore, the ultrafast SFMI developed here is likely to be suitable to observe the fast molecular dynamics occurring in living cells.
Here, we add a remark on the D micro s for Cy3-and goldlabeled molecules in the PM. The model of hop diffusion in the  (σ xy = single-molecule localization precision). Only in the typical MSD-Δt plot for a single 5xCy3-Tf molecule bound to TfR obtained at 45 kHz (a, bottom), 4σ xy 2 was not subtracted due to large errors in its estimation. Green curves are the best-fit functions for the hop-diffusion fitting (top and middle rows) and confineddiffusion fitting (bottom row for 45 kHz observations of 5xCy3-Tf). (B) Approximately 80% of Cy3-DOPE and TfR undergo suppressed diffusion in the intact apical PM, whereas >90% of them undergo simple-Brownian diffusion in the actin-depleted blebbed PM (shaded histograms in b), as revealed by the method for the statistical classification of each single-molecule trajectory into a suppressed-, simple-Brownian-, or directed-diffusion mode. (B a) The basic idea for the classification of trajectories into suppressed, simple-Brownian, and directed diffusion: the parameter RD (relative deviation) describes the extent to which the observed diffusion deviates from simple-Brownian diffusion at a time sufficiently later from time 0; i.e., the actual MSD divided by the calculated MSD from the short-term diffusion coefficient (D 2-4 ) assuming simple-Brownian diffusion. See Materials and methods. The RD value is <<, ≈, or >> 1, when the molecules are undergoing suppressed, simple-Brownian, or directed diffusion, respectively Kusumi et al., 1993;Murase et al., 2004;Suzuki et al., 2005). The suppressed-diffusion mode includes both the confined-diffusion and hop-diffusion modes. (B b) Classification of individual trajectories based on the RD histograms for simulated simple-Brownian particles (open bars; n = 5,000). The RD values giving the 2.5 percentiles of the particles from both ends of the distribution, referred to as RD min and RD MAX , were obtained (red and blue vertical lines, respectively). Each experimental single-molecule trajectory was classified into suppressed (confined and hop), simple-Brownian, or directed diffusion when its RD value was smaller than RD min , between RD min and RD MAX , and greater than RD MAX (no trajectory fell in this category), respectively. The distributions of RDs for the Cy3-DOPE and TfR trajectories are shown by shaded histograms (n = 50 and 20 for the intact and actin-depleted blebbed PM, respectively). (C a and b) In the blebbed apical PM, Cy3-DOPE (a) and Cy3-Tf bound to TfR (b) exhibited single diffusion coefficients that are ≈20× greater than D MACRO in the intact apical PM. The distributions of D micro (=D 2-4 ) are underestimated Fujiwara et al.
Journal of Cell Biology compartmentalized PM suggests that D micro is about the same as the diffusion coefficient of unhindered diffusion in the actindepleted PM, but in all the molecules and cells examined thus far, the D micro s in the intact PM were smaller than those for the unhindered diffusion coefficients in the actin-depleted PM ( Table 2; Fujiwara et al., 2002;Fujiwara et al., 2016;Murase et al., 2004;Suzuki et al., 2005). This is probably because the time resolutions of 25-100 µs achieved thus far (10-40 kHz) are not sufficient to observe trajectories near the compartment boundaries, thus effectively reducing the step sizes in the trajectory, which will reduce the diffusion coefficient within the compartment, D micro .
Direct determination of the dwell lifetime of each membrane molecule within each compartment using ultrafast SFMI We directly evaluated the dwell lifetime of each molecule in each compartment using ultrafast SFMI and developing an improved method for detecting the moment (instance) at which the observed molecule undergoes the hop movement across the compartment boundary. The moments of hops of the observed molecule can be found in its single-molecule trajectory by the method of detecting a Transient Increase of the effective Local Diffusion (TILD) in the trajectory (Fig. S5). TILDs are likely to occur when a molecule hops between two membrane compartments, but the analysis itself remains model independent. TILDs were detected in all of the experimental trajectories obtained at 0.1-and 0.167-ms resolutions (for Cy3-DOPE and TfR, respectively) in the intact PM (see The duration between two consecutive TILD events represents the dwell duration within a compartment. The distributions of the dwell durations are shown in Fig. 5 A. They could be fitted with single exponential decay functions, with lifetimes of 9.2 ± 0.34 ms for Cy3-DOPE and 23 ± 1.5 ms for TfR. The exponential shape of this distribution is consistent with the prediction by hop-diffusion theory developed here (see "Supplemental theory 1. Expected distribution of the dwell lifetimes: Development of the hop diffusion theory" in the Supplemental text). The decay time constants will be called the "dwell lifetimes" in the remaining part of this paper. The dwell lifetime for TfR is longer than that for Cy3-DOPE by a factor of 2.5, probably because the fence and picket effects both act on TfR, whereas only the picket effect works on Cy3-DOPE.
The average dwell lifetime within a compartment could also be evaluated from the median values of L and D MACRO determined by the hop-diffusion fitting of the MSD-Δt plot, using the equation τ = L 2 /4D MACRO (Fig. 4, C and D; and Table 2; Supplemental theory 1 in the Supplemental text). This provided dwell lifetimes of 8.5 ms for Cy3-DOPE and 24 ms for TfR, which are in good agreement with the values obtained based on the dwell lifetime distributions determined by the TILD method (Table 2).
In our previous studies using gold probes, we obtained L from ultrafast single-gold-particle tracking, but needed to separately obtain D MACRO using fluorescently tagged molecules (by observing them at video rate) to obtain the average dwell lifetime within a compartment (as L 2 /4 D MACRO ; Fujiwara et al., 2002Fujiwara et al., , 2016Murase et al., 2004). This is because gold probes induced crosslinking of the target molecules, elongating the dwell lifetime, and thus provided reduced D MACRO values. Using the ultrafast SFMI, we can directly determine the dwell lifetime of each molecule in each compartment (not just the averaged lifetime).
An anomaly analysis provided further proof of the hop diffusion of Cy3-DOPE and TfR in the compartmentalized PM Generally, the MSD can be expressed as a function of the time interval Δt, where 0 ≤α ≤2 (α = 1 for simple-Brownian diffusion, 0 ≤α <1 for suppressed anomalous diffusion, 1 <α <2 for super diffusion, and α = 2 for ballistic motion; Saxton, 1994Saxton, , 1996Feder et al., 1996;Simson et al., 1998;Fujiwara et al., 2002). This equation can be rewritten as This relationship is plotted in Fig. 5 B. A very broad time range of ∼5 orders of magnitude (from 0.044 ms to 2 s) was covered, which was made possible by the development of the ultrafast SFMI.
In this display, simple-Brownian type diffusion is represented by a flat line (since α = 1 in simple-Brownian cases, the slope α−1≈0; i.e., no time dependence) and suppressed diffusion is characterized by negative slopes ([α−1] <0; i.e., α <1). Namely, the level of the diffusion anomaly can be parameterized by the parameter α (Feder et al., 1996;Simson et al., 1998). In the actindepleted blebbed PM, both Cy3-DOPE and TfR exhibited flat lines in the entire time range of 0.3-30 ms, unequivocally demonstrating that they both undergo simple-Brownian diffusion in the blebbed PM.
On the contrary, in the intact apical PM, the plots for Cy3-DOPE and TfR (Cy3-Tf and 5xCy3-Tf) exhibited suppressed due to the insufficient time resolution for measuring D micro within a compartment, even at a 0.1-ms resolution. Arrowheads indicate the median values (summarized in Table 2). These diffusion coefficients in the blebbed PM are slightly smaller than those obtained with gold probes Fujiwara et al., 2016). This is probably due to the residual actin filaments in the T24 cells employed here, since the membrane-bound actin filament meshes are much denser in T24 cells than the NRK cells used previously. (D) Cy3-DOPE and TfR exhibited similar compartment size distributions, suggesting that the underlying mechanism for the compartmentalization would be the same for the phospholipid and the transmembrane protein. Arrowheads indicate the median values. The statistical test methods, parameters (number of experiments), and P values are summarized in Table 2.

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Journal of Cell Biology diffusion (negative slopes) in the time ranges of 0.5-30 and 0.13-70 ms, respectively, indicating that the suppressed diffusion of Cy3-DOPE and TfR was detectable in these time ranges. Meanwhile, in longer and shorter time ranges, the plots asymptotically approached flattened plots, consistent with simple-Brownian diffusion in these time regimes.
These results can readily be explained by the hop-diffusion model in the compartmentalized PM for both Cy3-DOPE and TfR. In the shorter time regime (<0.5 and 0.13 ms, respectively), Table 2. Examination of the ultrafast SFMI developed here to determine whether it can correctly evaluate hop-diffusion parameters for Cy3-DOPE and TfR in the apical PM, previously found by ultrafast gold-particle imaging (dwell lifetimes were previously determined using additional data from video-rate observations of fluorescently labeled molecules)

Mode of Motion (%)
Cmpt. Size (L, nm) b  c D MACRO s for Cy3-DOPE and Cy3-TfR were determined by the hop-diffusion fitting. D MACRO s for gold-DOPE and gold-TfR were generally underestimated, due to the crosslinking effect of the gold probes. Therefore, previously they were mostly determined by the observations of Cy3-DOPE and Cy3-TfR at 30 Hz (Fujiwara et al., 2016). D MACRO of gold-DOPE in NRK cells was determined for the diffusion in the time window of 30 ms (using the data obtained at 40 kHz). In the apical PM of the NRK cell, the PM exhibited nested double compartments, and the D MACRO of DOPE over smaller compartments could not be determined using fluorescent probes Fujiwara et al., 2016). d Dwell lifetime within a compartment (τ) for Cy3-labeled molecules was determined by the exponential fitting of the distributions of the residency duration for each molecule in each compartment, obtained by the TILD analysis (Fig. S5). SEM was determined from the fitting error of the 68.3% confidence interval. τ for gold-tagged molecules was calculated from the median values of L and D MACRO , using the equation τ L 2 /4D MACRO . e Gold-DOPE and gold-TfR data are from Fujiwara et al. (2002) Cy3-DOPE and TfR collide with the compartment boundaries less frequently, approaching simple-Brownian diffusion and thus providing the diffusion coefficient of D micro within a compartment (although this is still underestimated due to the collisions with the compartment boundaries. However, in this time regime, translocation across the compartment boundaries hardly occurs). In the longer time regime (>30 and 70 ms, respectively), the diffusion of Cy3-DOPE and TfR represents that among compartments, because the detailed shorter term behaviors are averaged out, and their movements thus resemble simple-Brownian diffusion again, with a diffusion coefficient of D MACRO . In the time ranges of 0.5-30 and 0.13-70 ms for Cy3-DOPE and TfR (Cy3-Tf and 5xCy3-Tf), respectively, since the effect of confinement within a compartment becomes increasingly evident with a lengthening of the observed time window until occasional hops across the compartment boundaries start alleviating this confinement, the MSD/(4t), which is the y-axis in this plot, is expected to decrease from D micro until it reaches D MACRO . The dashed curves in Fig. 5 B represent the results of the Monte Carlo simulations that resemble the experimental data. The cyan curve would be the best for representing the diffusion of TfR in the apical PM. In the longer time regime (>70 ms), it approached the experimental result for Cy3-Tf at 30 Hz, while in the shorter time regime (<0.8 ms), it approached the experimental data for 5xCy3-Tf at 45 kHz, and at its shorter limit, it approached the flat line of the Cy3-Tf observed in the blebbed PM.
TfR and Cy3-DOPE undergo hop diffusion in the basal PM, exhibiting virtually the same compartment sizes and dwell lifetimes as those in the apical PM Single molecules of TfR (TfR's N-terminus is located in the cytoplasm and it was fused to the Halo-tag protein and labeled with TMR) were observed in the basal PM outside the focal adhesion region marked with mGFP-paxillin (called bulk basal PM), at a frame rate of 6 kHz (0.167-ms resolution; 200 times faster than normal video rate) and compared with the results obtained in the apical PM ( Fig. 6 A and Video 5).
Using the MSD-Δt plot for each TfR trajectory, we found that the majorities of the trajectories were categorized into the suppressed diffusion mode in the basal PM, like those in the apical PM (72 and 80%, respectively; Fig. 6 B and Table 3). Consistently, the MSD-Δt plots of 81 and 82% of the trajectories obtained in the basal and apical PM, respectively, could be fitted by the hop-diffusion fitting equation, indicating that most TfR molecules undergo hop diffusion in the basal PM as well as in the apical PM. Interestingly, the median compartment size in the basal PM (109 nm) was quite comparable to that in the apical PM (103 nm; Fig. 6 C; non-significant difference; all statistical parameters and test results for diffusion parameters are summarized in Table 3).
The distributions of the TfR residency durations within a compartment in the basal PM, obtained by the TILD analysis, Cy3-DOPE and TfR (see the main text). Using the data obtained at time resolutions of 0.022 ms (45 kHz; only for TfR labeled with 5xCy3-Tf), 0.1 ms (10 kHz; for Cy3-DOPE), 0.167 ms (6 kHz; for TfR labeled with Cy3-Tf), and 33 ms (30 Hz; for Cy3-DOPE and TfR labeled with Cy3-Tf), the mean values of log(MSD/time) averaged over all trajectories were obtained and plotted as a function of log(time). The results of TfR using Cy3-Tf and 5xCy3-Tf were different in the time ranges shorter than 1 ms, due to the differences in the observation frame rates (6 and 45 kHz, respectively). During 0.167 ms, which is the frame time in 6-kHz observations, TfR still collides with the compartment boundaries, but this occurs much less often when the frame time is 0.022 ms (frame time in 45-kHz observations). Therefore, in shorter time ranges, the results obtained at 45 kHz (using 5xCy3-Tf) are better (see the simulation results shown by the dashed cyan curve). For the same reason, the 10-kHz data using Cy3-DOPE show that due to the insufficient time resolution, the pure simple-Brownian diffusion within a compartment could not be measured even at this frame rate. Dashed curves represent the results of the Monte Carlo simulations, resembling the experimental data (see Materials and methods for the simulation parameters). Note that the phospholipid probes are located in the PM outer leaflet, and yet they undergo hop diffusion. This is probably because, as proposed previously , the transmembrane proteins anchored to and aligned along the actin mesh (pickets; see Fig. 3 A) form the diffusion barrier in both the outer and inner leaflets of the PM. The picket effect is due not only to the steric hindrance of the picket proteins, but also to the hydrodynamic-friction-like effect from the surface of the immobilized picket proteins on the surrounding medium . Monte Carlo simulations showed that 20-30% occupancy of the compartment boundary by the immobile picket proteins (bound to the actin fence) is sufficient to cause confined + hop diffusion of the phospholipids in the PM outer leaflet . could be fitted by a single exponential decay function, as predicted by the hop-diffusion theory (Supplemental theory 1 in the Supplemental text), providing the dwell lifetimes of TfR in a compartment of 24 ± 1.6 ms in the basal PM as compared with 23 ± 1.5 ms in the apical PM (Fig. 6 D). Namely, between the apical PM and the basal PM of T24 cells, TfR dwell lifetimes within a compartment as well as the compartment sizes were very similar to each other, despite that the apical PM is facing the cell-culture medium whereas the basal PM is located very closely to the coverslip and responsible for the cell's binding to the coverslip. This was quite surprising to us, and so, the average residency time (τ) within a compartment was evaluated by another method, using the equation τ L 2 /4D MACRO . The median values of L were 109 and 103 nm (Fig. 6 C) and those of D MACRO were 0.12 and 0.11 µm 2 /s (Fig. 6 E) in the basal and apical PMs, respectively. These provided the average residency times of 25 and 24 ms in the basal and apical PM, respectively, in agreement with the values obtained by the TILD method. Therefore, we conclude that the TfR's dwell lifetimes in a compartment were virtually the same in both the basal and apical PMs.  Table 3.
The phospholipid, Cy3-labeled L-α-dioleoylphosphatidylethanolamine (Cy3-DOPE), also undergoes hop diffusion, exhibiting practically the same median compartment sizes as that for TfR in the basal PM and those for Cy3-DOPE and TfR in the apical PM (Fig. 2 C). The dwell lifetime of Cy3-DOPE within a compartment was shorter than that of TfR by a factor of ≈2.5 in both the basal and apical PMs (Fig. 6 D). Consistently, the D MACRO values of Cy3-DOPE were the same in the basal and apical PMs (Fig. 6 E). Overall, the bulk basal PM is compartmentalized in virtually the same manner as the apical PM, in T24 cells, with regard to the compartment sizes and temporary confinements of TfR and Cy3-DOPE.
Equality of the compartment size for TfR and Cy3-DOPE in the same membrane suggests that the underlying mechanisms for the compartmentalization are the same for these two molecules. This result is consistent with the model that the PM is compartmentalized by the actin-based fences and transmembrane protein pickets anchored to and aligned along the fences (Fig. 3 A).
Since the compartment size distributions of the apical and basal PMs are quite similar, the following possibility was examined: the hop diffusion detected by this camera might simply be an apparent phenomenon caused by the photo response non-uniformity (PRNU) of the developed camera system. PRNU might affect the single-molecule localization precisions through pixel-to-pixel variations in the sensitivity, which might make the molecules appear like those undergoing hop diffusion. The test result, shown in Fig. S1 of the companion paper (Fujiwara et al., 2023), indicated that PRNU of the developed camera is quite similar to that of EM-CCD camera and would scarcely affect the single-molecule localization precisions. Furthermore, as described in the companion paper, we found different compartment sizes within the FA region in the basal PM. Therefore, we conclude that PRNU would not induce apparent hop-like movements for membrane molecules in the PM.
Prolonged confinement of activated EGF receptor (EGFR) within a compartment EGFR exists as both monomers and dimers before EGF stimulation, and they interconvert rapidly with a dimer lifetime of  ≈13 s (Chung et al., 2010). After stimulation, EGFR tends to form rather stable dimers and greater oligomers for self-phosphorylation and activation (Chung et al., 2010;Low-Nam et al., 2011;Huang et al., 2016). We examined whether the EGF ligation of the receptor affects its lateral diffusion, because this will determine the rate of activated EGFR spreading along the PM by diffusion, which will provide a basis for a variety of mechanisms for EGFR signal propagation (Verveer et al., 2000;Reynolds et al., 2003;Koseska and Bastiaens, 2020). EGFR (fused to the Halo-tag protein at the cytoplasmic C-terminus of EGFR, labeled with TMR) was observed in the basal PM at the level of single molecules at a frame rate of 6 kHz (0.166 ms resolution; Video 6). Due to the presence of nonlabeled endogenous EGFR, even the fluorescent spots with the intensity of a single EGFR might represent EGFR dimers and greater oligomers. Virtually all of the fluorescent EGFR spots underwent hop diffusion both before and after ligation (Fig. 7, A and B; and Table 3). Interestingly, their median compartment sizes were the same before and after stimulation (106 nm), and quite comparable to the compartment sizes in the basal PM found by observing TfR (109 nm) and Cy3-DOPE (107 nm; nonsignificant differences; Table 3), supporting the PM compartmentalization for all PM-associated molecules.
The dwell lifetime of EGFR within a compartment was increased from 18 ± 1.2 ms before stimulation to 27 ± 1.5 ms during 2.5-5 min after stimulation (Fig. 7 C), which reduced D MACRO by ≈42% (Fig. 7 D). Since EGFR is likely to form stable dimers upon stimulation, the slowed macroscopic diffusion or prolonged dwell lifetime within a compartment could be induced by the lower hop probabilities of the larger diffusant; i.e., engaged EGFR dimers. TfR is also a single-pass transmembrane protein and exists as constitutive dimers and their dwell lifetime in the basal PM was 25 ms, quite comparable to the dwell lifetime of EGFR after stimulation, suggesting that engaged EGFR primarily exists as dimers. This result is consistent with the "oligomerizationinduced trapping" model proposed previously (Iino et al., 2001;Murakoshi et al., 2004;Heinemann et al., 2013). The longer confinement of engaged receptors within or near the compartment where the ligand EGF was originally received will be beneficial for localizing the signal, which might in turn induce the local reorganization of the actin cytoskeleton required for membrane ruffling and chemotaxis.

Discussion
The ultrahigh-speed camera system developed in this study has enabled the fastest SFMI ever performed. This would represent the ultimate rate possible with the currently available fluorescent molecules, e.g., 0.1 ms resolution with a 20-nm localization precision for single Cy3 molecules, using a saturating TIR illumination laser intensity of 79 µW/µm 2 , for a frame size of 256 × 256 pixels (14 × 14 µm 2 ). Under these conditions, ∼1.6% of the single Cy3 molecules could be tracked for periods longer than 100 frames (10 ms). Approximately 14% of single Cy3 molecules could be tracked for 100 frames under our standard conditions (oblique-angle illumination at a laser intensity of 23 µW/µm 2 with a 39-nm localization precision). When better fluorophores (faster emission and slower photobleaching) are developed, the time resolution for single-molecule imaging could be enhanced to 22 µs or more (for a frame size of 7.5 × 3 µm 2 in 128 × 64 pixels), without affecting other parameters ( Fig. S3 and Table 1). Presently, the fastest practical time resolution for SFMI is 33 µs with single-molecule localization precision of 34 nm for a frame size of 7.1 × 6.2 µm 2 (128 × 112 pixels).
For tracking just one molecule at a time, MINFLUX has recently been developed, and its performance on the glass is superb (Balzarotti et al., 2017;Eilers et al., 2018;Schmidt et al., 2021). It has achieved single-molecule localization precisions of 2.4 nm and <20 nm with 0.4-and 0.12-ms temporal resolutions, respectively. However, it can track only one molecule at a time (but could be expanded to track a few isolated molecules at a time). In contrast, the camera-based single-molecule imaging allows the observations of several tens to hundreds of molecules simultaneously in a field of view (like an entire cell). The camera-based method can thus be used to investigate the interactions and assemblies of molecules in live cells and study the different behaviors of molecules simultaneously in various regions of the cell, which is impossible with one molecule/cell technologies.
In live E. coli, MINFLUX provided a single-molecule localization precision of 48 nm with a 125-µs time resolution, under the conditions of observing the same molecule 742 times (Balzarotti et al., 2017). This is quite comparable to the results we obtained using the camera-based method, which allows us to simultaneously observe many single molecules at singlemolecule localization precisions of 20 and 34 nm (27 and 55 nm) at time resolutions of 100 and 33 µs using TIR (obliqueangle) illumination (Table 1). However, practically speaking, the same single molecules could be observed for 100-300 frames (10-30 ms at 10 kHz; localizing them 100-300 times), as compared with localizing one molecule 742 times with MIN-FLUX. Therefore, the ultrafast camera-based single-molecule imaging and MINFLUX provide complementary methods for examining the very fast dynamics of membrane molecules in living cells: MINFLUX is useful for tracking a single molecule for longer periods, whereas the ultrafast camera-based single-molecule imaging provides a method for observing many molecules at once to investigate their interactions and the location-dependent behaviors of single molecules in a cell. Single molecules labeled with two or three different colors could also be investigated readily using two or three cameras, respectively. The ultrafast camera system developed here was tested by examining whether it could detect the hop diffusion of Cy3tagged DOPE and TfR in the apical PM of live cells, as expected from the previous observations using gold-tagged molecules Fujiwara et al., 2016;Murase et al., 2004). We found that it can indeed detect the hop diffusion of Cy3tagged molecules (Fig. 3, 4, and 5; and Table 2) and the same compartment size as found by ultrafast gold-tracking (≈100 nm in the PM of T24 cells; Fig. 4 D and Table 2).
Ultrafast SFMI has brought forth new knowledge about the hop diffusion of membrane molecules. First, a new method to measure the dwell lifetime for each molecule in each compartment (not the overall average lifetime) was established. This now enables us to obtain the distribution of the dwell lifetimes for each visit of each molecule to each compartment. Furthermore, we developed a theory to describe the dwell lifetime distribution. The developed theory together with the experiments made possible with the ultrafast camera system has firmly established the method to describe the hop-diffusion characteristics of membrane molecules in the PM (Fig. 5 A), opening up the possibility that PM compartmentalization can be studied readily in various cell types.
Second, ultrafast SFMI allowed an anomaly analysis of a very broad time range of ≈5 orders of magnitude (from 0.044 ms to 2 s) for determining the diffusion properties of Cy3-tagged molecules. The anomaly analysis confirmed the hop diffusion of Cy3-DOPE and TfR in the PM and their simple-Brownian diffusion in the actin-depleted blebbed PM (Fig. 5 B).
Third, ultrafast SFMI revealed that not only the apical PM, which could be studied by single gold particle tracking before, but also the basal PM is compartmentalized, and that the hop diffusion and PM compartment properties in the basal PM are virtually the same as those in the basal PM (Fig. 6). In the apical and basal PMs, the compartment sizes are 102 and 108 nm, respectively (arithmetic averages for the values obtained for TfR and Cy3-DOPE); the dwell lifetimes of TfR in a compartment are 23 ± 1.5 and 24 ± 1.6 ms; and those of Cy3-DOPE are 9.2 ± 0.34 and 10 ± 0.61 ms. This answered the long-term question about whether the bulk basal PM is significantly different from the bulk apical PM. Even in non-polarized cells, the architecture of the PM facing the substrate (coverslip) within distances <40 nm (because gold particles of 40 nm in diameter do not enter the space between the basal PM and the coverslip) might be quite different from the PM facing just the cell culture medium. However, the results obtained in this work unequivocally demonstrated that the compartment organization and molecular hop diffusion in the basal PM are very similar to those in the apical PM.
Fourth, ultrafast SFMI showed that EGFR before and after stimulation undergoes hop diffusion, supporting the compartmentalization of the basal PM (Fig. 7). The engaged EGFR is confined in a compartment ≈2.5× longer than the non-engaged EGFR, which might be useful for inducing a location-dependent downstream signal.
Meanwhile, the simple-Brownian diffusion of both Cy3-DOPE and TfR in the actin-depleted blebbed PM indicates that actin filaments (and their associated transmembrane picket Figure 7. EGFR and ligand-engaged EGFR in the basal PM detected virtually the same compartment sizes as those found with TfR and Cy3-DOPE, supporting the PM compartmentalization, and the dwell lifetime of the engaged EGFR was longer than that of non-engaged EGFR (same compartment sizes). Following the stimulation with 10 nM EGF, microscope observations were performed between 2.5 and 5 min after the EGF addition.  Table 3.

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Journal of Cell Biology proteins) are responsible for compartmentalizing the PM (Fig. 3  A). This conclusion is consistent with previous observations, including the changes of the compartment sizes after the treatments with actin-modulating chemicals, latrunculin, cytochalasin D, and jasplakinolide Fujiwara et al., 2016;Murase et al., 2004), the equality of the compartment sizes determined by the lipid diffusion data and the actin mesh sizes on the apical PM cytoplasmic surface determined by electron tomography (Morone et al., 2006) and super-resolution microscopy (Xia et al., 2019;Garlick et al., 2022), and the results of single-molecule optical trapping (Sako and Kusumi, 1995;Sako et al., 1998). We believe that the PM compartmentalization is fundamentally important for understanding the PM function. For instance, PM compartmentalization would permit the cells to locally enhance phosphorylation at particular places on the PM, by sequestering the kinases in the compartment so that the local concentrations of kinases there exceed those of phosphatases, which are generally more abundant in the cytoplasm and have higher enzymatic activities than kinases; Kalay et al., 2012;Kusumi et al., 2012a). Compartmentalization could also confine stimulation-induced stabilized raft domains Kusumi et al., 2011;Kusumi et al., 2012a;Kusumi et al., 2012b;Suzuki et al., 2012;Kusumi et al., 2020) and engaged receptor oligomers (oligomerization-induced trapping; Iino et al., 2001). Nevertheless, only a few researchers have directly investigated and identified hop diffusion in the PM due to the difficulties in using large gold probes and visualizing them at high frame rates. The developed ultrafast camera system with single fluorescent-molecule sensitivities allows the researchers to investigate PM compartmentalization and hop/confined diffusion of membrane molecules quite readily, enabling faster progress of PM structure, dynamics, and function. Furthermore, simultaneous two-color ultrafast SFMI imaging is also possible (Koyama-Honda et al., 2020).
We believe that the developed ultrafast camera has the immediate value and broad utility to the cell biology community by the following reasons. The highest time resolutions in SFMI of many single molecules at once are now available to the cell biology community. Ultrafast single-molecule imaging method is now enhanced by the development of the theoretical framework for the analysis of single-molecule trajectories in the PM, which provides the simple means to examine the effects of PM compartmentalization on biological processes in/on the PM (previously, very special single gold-particle imaging and tracking were necessary) and to elucidate the principles governing the PM organization. The ultrafast camera will facilitate monitoring very fast molecular interactions, which have probably been missed in cell biology research.
Furthermore, as shown in the companion paper, the newly developed ultrafast camera system also offers an unprecedented opportunity to simultaneously perform the fastest data acquisition (1 kHz) of PALM (using the mEos3.2 probe with a 29-nm localization precision) and dSTORM (using the HMSiR probe with a 19-nm localization precision) in a view-field as large as 640 × 640 pixels (35.3 × 35.3 µm 2 with the pixel size of 55.1 nm), together with ultrafast SFMI. The data acquisition of 333-10,000 frames requires merely 0.33-10 s, enabling live-cell PALM. Therefore, we conclude that the ultrafast camera system developed in the present work holds enormous potential as an invaluable tool for the field of cell biology.

Materials and methods
Ultrahigh-speed intensified CMOS camera system: Design and operation Detailed description of the camera system ( Figs. 1 and 2; and Figs. S1, S2, and S3; and Table 1). See the schematic diagram of the developed camera system shown in Fig. 1 A. The new ultrahigh-speed camera system consists of the following three major components.
(a) A Hamamatsu image intensifier (V8070U-74), comprising a third-generation GaAsP photocathode with a quantum efficiency of 40% at 570 nm (Fig. 1, A a α), a three-stage microchannel plate allowing maximal gain over 10 6 , in which a non-linear noise increase was virtually undetectable (Fig. 1, A a β), and a P46 phosphor screen with a decay time of ∼0.3 µs from 90 to 10% (Fig. 1, A a γ). (b) A CMOS sensor (unless otherwise specified, we report the results obtained with the sensor developed for a Photron 1024PCI camera, but for some tests, we additionally report the results obtained with the newer sensor developed for a Photron SA1 camera), with a global shutter exposure was used ( Table 1). The 1024PCI (SA1) sensor is composed of 1,024 × 1,024 pixels with a 17 × 17-µm (20 × 20-µm) unit pixel size and operates up to 1 kHz or every 1 ms (5.4 kHz or every 0.19 ms) for the full-frame readout. The 1024PCI sensor can be operated at 10 kHz (0.1-ms resolution) with a frame size of 256 × 256 pixels, at 45 kHz (0.022-ms resolution) with a frame size of 128 × 64 pixels, and at 110 kHz (0.009ms resolution) with a frame size of 128 × 16 pixels. The SA1 sensor can also be operated at 16 kHz (0.063-ms resolution) with a frame size of 512 × 512 pixels, and at 45 kHz (0.022-ms resolution) with a frame size of 256 × 256 pixels (Table 1).
However, the actual observation frame rates for singlemolecule tracking with reasonable single-molecule localization precisions (say ≤50 nm) were limited to 10-30 kHz. This is because the number of fluorescent photons that can be emitted from a single fluorophore during <0.033-0.1 ms (the integration time of a camera frame); i.e., the frequency that a single fluorophore can be excited, is limited, as quantitatively described in "Estimation of the number of photons that can be emitted by a single Cy3 molecule during 0.1 ms: Triplet bottleneck saturation" in the Materials and methods (also see Fig. 1, E and F; and Fig. S3). Namely, with the development of our ultrafast camera system, the instrument is no longer the limitation for achieving higher time resolutions (faster than 10-30 kHz) for SFMI, and now the availability of fluorescent dyes has become the limitation. With the future development of new dyes with faster excitation and emission, without making a transition to the triplet state, observations even at 110 kHz would be possible. Indeed, we showed that when an average (mean) of five Cy3 molecules were attached to a single Tf protein (5xCy3-Tf), it can be observed at a rate of 45 kHz (Fig. 1 I; Table 1).
The readout noise, including the dark noise, of the 1024PCI and SA1 sensors at the full-speed readout at 21°C was 37 rootmean-square electrons/pixel, as measured by the manufacturer (Photron), and provided dynamic ranges of ≈800 and ≈1,200, respectively. The quantum efficiency of the CMOS sensor employed here was 40% (for both sensors), but since the CMOS sensor is placed after the image intensifier, the lower quantum efficiency of the CMOS sensor is not a critical issue in the developed camera system (the quantum efficiency of the image intensifier photocathode matters).
(c) A straight (1:1) optical-fiber bundle directly adhered to the phosphor screen of the image intensifier on one side (Fig. 1, A a γ, input side) and to the CMOS sensor on the other side ( Fig. 1, A b, output side). This enhanced the signal reaching the CMOS sensor by a factor of 5-10, as compared with the lens coupling used for our standard camera system employed for SFMI Kinoshita et al., 2017).
The electrons generated at the CMOS sensor were transferred, amplified (Fig. 1, A d), and digitized ( Fig. 1, A e) in 64 and 8 parallel paths, respectively, and then transferred to the host PC.
Basic design concept of the camera system (a) The use of a CMOS sensor rather than an sCMOS sensor or an EM-CCD sensor.
To achieve ultrafast SFMI, we used a CMOS sensor rather than an sCMOS sensor or an EM-CCD sensor, which are broadly employed in fluorescence microscopy. This is because the maximum frame rates attainable with the sCMOS sensors and EM-CCD sensors (typically 1.2-4 ms/frame for a 256 × 256-pixel image) are at least 10 times slower than those of the CMOS sensors employed here (0.1 and 0.022 ms/frame for a 256 × 256pixel image, using the Photron 1024PCI sensor and SA1 sensor, respectively; no binning). To avoid image distortions and achieve higher rates, we employed CMOS sensors with a global shutter (most sCMOS cameras employ rolling shutters; although some newer sCMOS cameras are equipped with global shutters, their frame rates are slower than those when rolling shutters are employed).

(b) The use of an image intensifier (also see the beginning of
Results in the main text).
An obvious problem when using the CMOS camera is its high readout noise (30-40 vs. <5 root-mean-square electrons/pixel/ frame for CMOS vs. sCMOS; the term "readout noise" used in this report always includes the dark noise of the sensor). As described in the previous section, the readout noise of the 1024PCI and SA1 sensors employed in this study was 37 rootmean-square electrons/pixel/frame (at the full-speed readout at 21°C). Meanwhile, the background noise signal (b in the equation in the caption to Fig. S2) was 0.035 ± 0.058 detected photons/ pixel/frame on the glass using the oblique illumination (mean ± SD; in the case of TIR illumination, b = 0.038 ± 0.059 detected photons/pixel/frame) and 0.068 ± 0.17 detected photons/pixel/ frame on the cellular apical PM (oblique illumination). Therefore, the background noise signal is much smaller than the readout noise, as expected (i.e., N p << Nr as described in the subsection "Basic conceptual strategy to address the large readout noise of the CMOS sensor" in the main text). As shown in Eq. 3 there, G should be much greater than Nr/N p , i.e., G >> Nr/N p . Therefore, G>>37/0.035 (or 0.068) or G>>1,100 (or 540) for single fluorescent molecules on the glass (apical PM). As described in the following paragraphs, by also considering S p , we found a total amplification gain G of 8,100 useful for SFMI.
Next, we address the signals from the fluorophores. Consider our standard test conditions for observing a single Cy3 molecule, using a frame rate of 10 kHz and oblique illumination for excitation at 23 µW/µm 2 (i.e., instrumentally less optimal as compared with the TIR illumination, but necessary for observing molecules in the apical PM as well as in the basal PM). Under these conditions, we obtained 34 ± 2.4 detected photons/molecule/frame ( Fig. 1 E top; including a quantum efficiency of the image intensifier photocathode of 0.4 at 570 nm; i.e., the number of photons that arrived at the photocathode of the intensifier was 85 ± 6.0 photons/molecule/frame), which realized a single Cy3-molecule localization precision of 38 ± 1.7 nm on the coverslip ( Fig. 1 F top). Namely, an average (total number of) of 34 ± 2.4 photons is detected on the CMOS sensor in the twodimensional Gaussian image of a standard deviation (SD; radius) of 123 ± 1.1 nm (2.2 ± 0.020 pixels × 55.1 nm/pixel) on the sample plane. (This image size was determined by the Gaussian fitting of each image for 50 Cy3 molecules immobilized on the glass under the TIR illumination at 79 µW/µm 2 ; this higher illumination was employed to determine the Gaussian image size more precisely than that obtainable under the oblique illumination at 23 µW/µm 2 , by generating ∼3 times more detected photons; see Consider how the 34 photons are distributed in the pixelized Gaussian image; i.e., the 2-dimensional intensity profile (pointspread function) of a single-molecule emitter pixelized by the CMOS sensor. The expected number of photons in the pixel located at the integer position (m,n), I(m,n), is given as (Huang et al., 2011): Here I total is the total number of detected photons in the image; i.e., 34 photons, and E m and E n are where erf represents a Gaussian error function, (m 0 ,n 0 ) is the sub-pixel emitter position, and s is the SD of the Gaussian spot profile; i.e., 2.2 ± 0.020 pixels. Using these equations, the peak pixel intensity I(0,0) (the number of detected photons at (m,n) =(0,0)) when a single-molecule emitter is located exactly at the center of the pixel (0,0) is estimated to be only 1.1 photons.
Under such low signal conditions, in which the pixel intensities within the Gaussian spot profile fluctuate frame by frame at the level of single photons, the photoelectrons emitted at the photocathode of the image intensifier should be amplified until the readout noise of 37 root-mean-square electrons/pixel/frame, which also fluctuates spatiotemporally, becomes negligible. To satisfy this condition, the photon amplification gain of the image intensifier of 33,200 (the ratio of the number of photons emitted from the phosphor screen of the image intensifier vs. the number of detected photons) was used throughout this study (see the following estimates for details). The image intensifier was coupled to the CMOS sensor by an optical fiber bundle. Therefore, due to the 39% loss (61% coupling efficiency) by the optical fiber bundle and the quantum efficiency of the CMOS sensor of 40%, the overall electron amplification (after photon detection) was a factor of 8,100 (33,200 × 0.61 × 0.4). Since the quantum yield of the photocathode of the image intensifier was 0.4, the total amplification of the incident photon of this camera system was 3,240 (one photon that arrived on the photocathode of the image intensifier generated an average of 3,240 electrons in the CMOS chip).
The overall electron amplification of 8,100× (after photon detection) was selected based on the following two considerations. First, the probability that the electron signal at the CMOS sensor amplified from a single detected photon (at the photocathode of the image intensifier) becomes greater than the readout noise should be sufficiently high. As shown in Fig. S1 E, the probability of detecting a single photon (as normalized by the number of detected photons/frame at the saturating levels of electron amplification) was increased to 90.0% at an overall electron amplification of 8,100×.
Second, the upper limit of the overall electron amplification is given by the full-well capacity of 30,000 electrons for the individual pixel (for the 1024PCI sensor employed for most experiments in this study). Considering the stochastic fluctuation of the signal, we set the electron amplification of the image intensifier so that the average signal in the peak pixel of 1.1 electrons/pixel/frame was amplified to <1/3 of the full well capacity (<10,000 electrons); i.e., amplification by a factor of <9,090 (= 10,000/1.1). Combining these two considerations, we employed an overall electron amplification by a factor of 8,100 (33,200 at the image intensifier), 10% less than the factor 9,090, to further reduce the possibility of pixel intensity saturation. (For detecting dimers and larger oligomers, the image intensifier gain should be reduced, although the probability of missing monomers would increase slightly.) (c) The design and conditions for visualizing and tracking single molecules for many frames (Fig. 2 C) Note that these illumination and amplification conditions were selected so that we could track single molecules for many frames in the apical PM (e.g., for longer than 99 frames for 14% of the Cy3 molecules immobilized on the glass, under the conditions of three-frame gap closing, as shown in Fig. 2 C). When only much shorter tracking is required (such as a single frame or in the case of PALM), much higher excitation laser power density could be employed for better single-molecule localization precisions. The best single-molecule localization precision we achieved with this camera system was 2.6 ± 0.099 nm (mean ± SEM), with a maximum number of detected photons/molecule/ frame of 11,400 ± 700 (mean ± SEM; n = 50 Cy3 molecules; Fig. 1, G and H; and Fig. S2 D), using a TIR illumination power density at 79 µW/µm 2 and a signal integration time of 16.7 ms. This is because, under these conditions, most Cy3 molecules become photobleached within a single frame (16.7 ms; i.e., virtually all of the photons possibly obtained from single molecules are concentrated in a single frame) due to the use of a TIR illumination density of 79 µW/µm 2 , by which the photon emission rate from a single Cy3 molecule is saturated (Fig. 1 E).
Estimation of the number of photons that can be emitted by a single Cy3 molecule during 0.1 ms: Triplet bottleneck saturation (Fig. 1, E and F;and Figs. S2 and S3) In ultrafast SFMI, the number of photons emitted from a single molecule during the duration of a single camera frame is the key limiting factor. This critically depends on how fast a molecule returns to the ground state from the excited state. In a conventional three-state model (ground, singlet-excited, and triplet-excited states), the maximum photon emission rate, k em (photons/s), can be described as (Schmidt et al., 1995): where τ s and τ T represent the lifetimes of the singlet-and triplet-excited states, respectively, k isc (= Φ T /τ s ) is the intersystem crossing rate, Φ is the fluorescence quantum yield, and Φ T is the quantum yield for triplet formation. Namely, the maximum photon emission rate, k em , critically depends on the triplet yield and lifetime, and hence this phenomenon is known as "triplet bottleneck saturation." For Cy3, τ s ≈ 1 ns, τ T ≈ 1 µs (in a specimen equilibrated with atmospheric molecular oxygen, the triplet lifetime is dominated by the collision rate of molecular oxygen with the fluorophore, which is about 10 6 /s; Kusumi et al., 1982), Φ = 0.15, and Φ T = 0.003 were employed (Sanborn et al., 2007;Dempsey et al., 2011). These values provide a k em of 3.8 × 10 7 photons/s. Therefore, the maximum number of photons available to form an image of a single Cy3 molecule, during the integration duration of 0.1 ms for each frame, is ∼3,800. Although this number would vary, depending on the spectroscopic parameters in the given environment, it provides a baseline for understanding such fast imaging. Assuming that 5-10% of the emitted photons will reach the intensifier photocathode (Rasnik et al., 2007;Gould et al., 2009), 190-380 photons will reach the photocathode of the image intensifier and be detected with a ≈40% quantum efficiency of the GaAsP photocathode at the Cy3 peak emission wavelength of 570 nm, providing the maximum number of photons detected during a 0.1-ms camera frame time of 80-150.
This estimate agrees well with the experimental results for evaluating the maximal photon emission rate of Cy3 by TIR illumination (here, since we are discussing saturation conditions, we focus on the results obtained by the TIR illumination rather than the oblique illumination) shown in Fig. 1 E top and Fig. S2 A left (the data displayed in Fig. S2 A left show the relationship of the single-molecule localization error with the number of detected photons/spot/frame obtained at a frame rate of 10 kHz, which was found to be consistent with the theory developed previously; Mortensen et al., 2010). In Fig. 1, E and F (top), with an increase of the illumination intensity, both the number of detected photons/molecule/frame and the single-molecule localization precision were enhanced. However, the extents of the enhancement were saturated, because the number of photons that a single molecule can emit during the 0.1-ms frame time was limited. Under the near-saturation conditions at a laser power density of 79 µW/µm 2 at the sample plane, the number of detected photons was ≈100 photons/frame ( Fig. 1 E top). This confirmed that the experimental result is consistent with the estimate of the maximum photon emission rate based on the three-state model (ground, singlet-excited, and triplet-excited states; 80-150 photons during 0.1 ms) as described in the previous paragraph, indicating that the saturation of Cy3 photoemission can be explained by the "triplet bottleneck saturation." Under the Cy3 saturation conditions, the improvement of the single-molecule localization precisions with an increase of the laser excitation intensity was limited to (saturated at) ≈20 nm for recordings at 10 kHz ( Fig. 1 F top and Fig. S2 A left).
Among the eight dyes examined here, JF549, TMR, JF646, Atto647N, SeTau647, and Cy5 exhibited saturation more readily than Cy3 and Alexa555 (Fig. S2 B). Furthermore, the average numbers of detected photons from single molecules during the 0.1-ms frame time obtainable at saturation were smaller than those of Cy3 and Alexa555, and thus their single-molecule localization precisions at 10 kHz at saturation were worse than those of Cy3 and Alexa555 (Fig. S3). At saturation, Cy3 provided slightly better single-molecule localization precision at 10 kHz than Alexa555 (Fig. S3 B). Therefore, we primarily employed Cy3 throughout the remaining part of this report. Note that this result depended not only on the photophysical properties of individual dyes, but also on the wavelength dependence of the photocathode quantum yield of the image intensifier. The quantum yield is lower (≈0.2) for the near-IR dyes, JF646, At-to647N, SeTau647, and Cy5, as compared with that (≈0.4) for the Cy3, Alexa555, JF549, and TMR dyes.
Determination of the number of detected photons/molecule/ frame (N) (Fig. 1, E and G;and Fig. 2,A and B;and Figs. S2 and S3) All of the fluorescent dye molecules used for determining the number of detected photons/molecule/frame were covalently bound to the coverslip coated with 3-APS, and 5xCy3-Tf was adsorbed on the coverslip coated with poly-D-lysine (see the previous subsection). The number of detected photons/molecule/frame was evaluated by multiplying the pixel intensity and the gain conversion factor of the developed camera system. This factor was determined by detecting the same number of photons on both the developed camera system and an Andor iXon DU-897 back-illuminated EM-CCD camera, with the known gain conversion factor of 0.03622 photons/pixel count at an EM gain of 50 and an imaging depth of 16 bits (measured by Zeiss for an ELYRA P.1 PALM system). Briefly, the images of fluorescent beads immobilized on a coverslip were split by a 50/50 beam splitter, and each image was projected onto the new camera system and the EM-CCD camera at the same time. The number of detected photons from each fluorescent bead/frame was obtained by multiplying the gain conversion factor of the EM-CCD camera and the entire pixel intensities of the spot on the EM-CCD camera. The number of photons that reached the EM-CCD camera was then obtained by dividing it by the quantum efficiency of the EM-CCD camera (0.95). The same number of photons must have reached the photoelectric cathode of the developed camera, and therefore, by multiplying with the quantum efficiency of the developed camera (0.4), the number of detected photons/frame/spot was obtained. By comparing this number with the entire pixel count of the fluorescent spot, the conversion factor of the developed camera system was evaluated. At the typical overall electron amplification of 8,100× employed for the observations at a frame rate of 10 kHz (0.1-ms frame time) in the present research, the gain conversion factor of the developed camera system was estimated to be 0.00363 (0.00136) photons/pixel count in an imaging depth of 10 (12) bits for the 1024PCI (SA1) sensor.
Detecting single-molecule fluorescence spots, determining their positions in the image, and linking the positions to generate trajectories Each and every fluorescent spot in an image was detected and its position was determined, using an in-house computer program based on a spatial cross-correlation matrix (Gelles et al., 1988;Fujiwara et al., 2002;Fujiwara et al., 2016). The image (signal intensity profile) was correlated with a symmetric 2-dimensional Gaussian point spread function (PSF) with a standard deviation of 123 nm (kernel; 123 nm was used for Cy3 and the value was individually determined for each fluorescent probe; see the caption to Fig. S2; Mashanov and Molloy, 2007). The "maximum entropy" thresholding (Sahoo et al., 1988) was applied to the cross-correlation matrix. The spot representing a single molecule was detected as that containing an area with a size ≥5 pixels, which was defined as the size of the area where the values of the cross-correlation matrix were greater than or equal to the threshold value (a constant value for the whole image). The spot position (x, y coordinates) was determined in the following way. First, the centroid of the cross-correlation matrix was calculated, and then an 830-nm-diameter circular region (55.1 nm/pixel x 15 pixels) with its center placed at the centroid was determined as the region that includes the intensity profile of the fluorescent spot. Second, the spot position was determined by fitting the intensity profile in the circular region to a symmetric 2-dimensional Gaussian PSF, using the following formula: where the fitting parameters were A (amplitude), (x 0 ,y 0 ) (the sub-pixel center position of the spot), and s xy (standard deviation of the symmetric Gaussian PSF), and the value for B (background intensity offset) was measured (background signal intensity divided by the number of pixels). To generate trajectories from the determined coordinates of the fluorescent spots in an image sequence, the positions were linked when the displacement between the spots in the consecutive frames was smaller than the distance giving the ≥99.7% cumulative probability for a particle undergoing simple Brownian diffusion with an expected diffusion coefficient and a position localization error (Schütz et al., 1997;Sahl et al., 2010).
Observing the time series of the number of detected photons/ 0.1-ms-frame from single Cy3 molecules, determining the signal-to-noise ratios (SNRs) of single Cy3 molecule images, and evaluating the durations in which single Cy3 molecules could be consecutively observed (the bright/dark periods [on/ off-periods] in the images of single Cy3 molecules and the gap closing for short off-periods; Fig. 2) Time-dependent changes in the number of detected photons/ 0.1-ms-frame from single Cy3 molecules immobilized on a coverslip were observed at 10 kHz, under our standard obliqueangle laser illumination conditions of 23 µW/µm 2 (Fig. 2 A). The molecules that were observable in the first frame of the image sequence were selected. The number of detected photons during a single image frame in an 830 nm diameter (55.1 nm/ pixel × 15 pixels) circular region including a fluorescent spot, called the "spot intensity," was measured and plotted against the frame number at 10 kHz. To evaluate the signal intensity in the background, the number of detected photons during a single image frame in an adjacent region with the same shape and size, called the "BG signal intensity" was measured. On-and offperiods were determined based on the thresholding employed for the spot detection.
The signal-to-noise ratio (SNR) for the image of a single molecule was determined in the following way (Fig. 2 B). From the time-dependent fluorescence signal intensities, the histograms of the numbers of detected photons/0.1-ms for the images of single molecules; i.e., the histograms for the "spot signal intensities" and the "background [BG] signal intensities" during the on-periods, were obtained, and their mean signal intensities (I Spot and I BG , respectively) and standard deviations (σ Spot and σ BG , respectively) were calculated. The SNR was evaluated using the following equation: The mean SNR of single Cy3 molecules immobilized on the coverslip was 2.5 ± 0.11 for 50 Cy3 molecules with on-periods of 15-150 frames at 10 kHz, whereas the mean SNR on the apical PM was 1.8 ± 0.056 for 50 Cy3-DOPE molecules with on-periods of 1,000 frames at 10 kHz, as obtained under our standard oblique-angle laser illumination conditions of 23 µW/µm 2 .
The distributions of the durations of the on-periods and those after neglecting the off-periods lasting for 1, 2, or 3 frames (called "gap closing") are shown in Fig. 2 C. The off-periods are induced by the occurrences of non-emission periods of a single dye molecule (which could induce no or dim images in a frame, depending on the duration of the non-emitting period; the offperiods tend to occur more often during the observations of living cells, probably due to the existence of various chemicals in the cell and the culture medium). The off-periods also occur when the signal from a single molecule is missed for short periods, due to the fluctuations of the fluorophore signal intensity (I Spot ) as well as the higher background (I BG ) and its fluctuation (σ BG ). Therefore, gap closing for 1-3 frames is often employed in single-molecule tracking studies. For describing the results obtained in living cells in the present report, we used 3-frame gap closing.
Cell culture Human T24 epithelial cells (the same as the ECV304 cell line used in the previous research [Murase et al., 2004], which turned out to be a sub-clone of T24; Dirks et al., 1999) were grown in Ham's F12 medium (Sigma-Aldrich) supplemented with 10% fetal bovine serum (Sigma-Aldrich). Cells were cultured on 12-mm diameter glass-bottom dishes (IWAKI), and single-molecule observations were performed two days after inoculation. For culturing cells expressing mGFP-paxillin, the glass surface was coated with fibronectin by an incubation with 5 µg/ml fibronectin (Sigma-Aldrich) in phosphate-buffered saline (PBS; pH 7.4) at 37°C for 3 h.
To form PM blebs (up to 20 µm in diameter) depleted of the actin-based membrane-skeleton, the cells were incubated with 1 mM menadione for 5 h at 37°C , and then treated with 100 nM latrunculin A (a gift from G. Marriott, University of California, Berkeley) on the microscope stage for 5 min at 37°C. All measurements were performed within 5-15 min after the addition of latrunculin A Fujiwara et al., 2016).
Examination of the photo-induced damage to the cell during ultrafast SFMI (Fig. 2, D and E) Cell viability was assessed by staining with 1 µM TOTO-3 iodide (Molecular Probes), which only stains dead cells (Zuliani et al., 2003), at 37°C for 5 min and then observing the stained cells using epi-illumination with a 594-nm laser. To obtain reference images of dead cells, the cells were treated with 100 µM H 2 O 2 at 37°C for 1 h. The fluorescence intensity of TOTO-3 in the 5.5 × 5.5-µm area inside the nucleus was measured, and the histograms of signal intensities were obtained. Based on the negative and positive controls (Fig. 2, D and E), a threshold fluorescence intensity of 4.0 × 10 5 (arbitrary unit = AU) was selected to categorize the live and dead cells (96% [90%] of the negative [positive] control cells were categorized as alive [dead]).
Note the following. Once the desired single-molecule localization precision is chosen for each experiment, the required total number of photons emitted from the molecule during a frame to achieve the precision will be the same, irrespective of the frame rate (inverse of the frame time). Therefore, the total number of photons elicited by laser excitation during a frame time will be the same, regardless of whether the observations are made at video rate (33-ms frame time) or at 10 kHz (0.1-ms frame time, if photon emission saturation does not occur). Namely, the direct photo-damage to the cell per frame due to the Fujiwara et al.
Journal of Cell Biology excitation laser illumination is expected to remain constant even when the laser illumination intensity and the frame rate are increased, as long as the single-molecule localization precision required for the experiment and the number of observed frames (not the total observation duration) are the same (in the absence of photon emission saturation). However, direct experiments are required to test the phototoxicity to cells, as shown here. This is partly because slight photon-emission saturation occurred under our standard illumination conditions of 23 µW/µm 2 (Fig. 1 E), and also because higher illumination photon densities could induce two-photon events and enhance the secondary reactions of the photo-induced molecules produced at high densities in the cell, which could generate cytotoxic substances.
Preparation of Cy3-DOPE and Cy3-Tf and cell surface labeling Cy3-DOPE was prepared and incorporated in the PM as described previously Murase et al., 2004), except that the final concentration of Cy3-DOPE added to the cells was 10 nM. Cy3-Tf used for the measurements at 6 kHz (0.167-ms resolution) was generated by incubating 6.5 µM monofunctional sulfo-Cy3 (GE Healthcare) with 6.3 µM human holo Tf (Sigma-Aldrich) in 0.1 M carbonate buffer (1 ml, pH 9.0) at 25°C for 60 min, followed by desalting column chromatography (PD-10, GE Healthcare, equilibrated and eluted with PBS) to remove the unreacted dye. The dye/protein molar ratio of the Cy3-Tf was 0.2. At this ratio, more than 90% of the fluorescent spots in the image are expected to represent single Cy3 molecules, based on the Poisson distribution, which was confirmed by single-step photobleaching of the fluorescent spots of Cy3-Tf (as well as Cy3-DOPE). Tf conjugated with multiple Cy3 molecules, used for the measurement at 45 kHz (0.022-ms resolution), was generated by incubating 1.25 mM monofunctional sulfo-Cy3 with 6.3 µM human holo Tf (mixed at a molar ratio of ∼200:1), which provided Cy3-Tf with the dye/protein molar ratio of ≈5 (called 5xCy3-Tf).
To label TfR in the PM with Cy3-Tf, first the Tf (originated from FBS) already bound to TfR on the cell surface was partially removed by incubating the cells in 1 ml Hanks' balanced salt solution buffered with 2 mM TES (Dojindo), at pH 7.4 (HT medium) and 37°C for 10 min, and then after washing, HT medium containing 10 nM Cy3-Tf (or 5xCy3-Tf) was added to the cells at a final concentration of 0.5 nM. At this concentration, single molecules of Cy3-Tf bound to the apical PM could be observed without replacing the medium with the Cy3-Tf-free medium.
The observations were completed within 10 min after the Cy3-Tf addition.

Halo-TfR and mGFP-paxillin expression in T24 cells and TMR labeling of Halo-TfR
The cDNA encoding human TfR (GenBank: M11507.1) fused to the Halo-tag protein at the TfR's N-terminus (Halo-TfR) was generated by replacing the cDNA encoding the EGFP protein, in the EGFP-TfR plasmid, with that of the Halo-tag protein (Promega), with the insertion of a 45-base linker (15 amino acids, with the sequence SGGGG ×3) between Halo and TfR. The cDNA encoding human paxillin isoform alpha (NCBI reference sequence: NM_002859.3; cloned from the human WI-38 cell line) fused to mGFP at the N-terminus was subcloned into the EBV-based episomal vector, pOsTet15T3 (gift from Y. Miwa, University of Tsukuba, Japan), which bears the tetracyclineregulated expression units, the transactivator (rtTA2-M2), and the TetO sequence (a Tet-on vector; Shibata et al., 2012;Shibata et al., 2013). To avoid perturbing the FA structure by overexpressing mGFP-paxillin, doxycycline induction was not employed, but leaky expression was used. However, due caution is required when interpreting how effectively mGFP-paxillin imitates the behaviors of native paxillin.
T24 cells were transfected with the cDNA encoding mGFPpaxillin, using the Lipofectamine LTX and Plus reagents (Invitrogen) and with the cDNA encoding Halo-TfR using Nucleofector 2b (Lonza), following the manufacturers' recommendations. To covalently link TMR to Halo-TfR, T24 cells coexpressing Halo-TfR and mGFP-paxillin were incubated in HT medium, containing 10 nM TMR-conjugated Halo-ligand (Promega) at 37°C for 1 h, and then washed three times with HT medium. The remaining unbound ligand in the cytoplasm was removed by incubating the cells in HT medium for 30 min, and then washing the cells three times with HT medium.

EGFR-Halo expression in T24 cells and its TMR labeling
The cDNA encoding human EGFR (GenBank: X00663.1) tagged with Halo (EGFR-Halo) was generated by replacing the cDNA encoding the YFP protein, in the EGFR-YFP plasmid (a gift from I. R. Nabi, University of British Columbia, Canada), with that of the Halo 7-tag protein (Promega), with the insertion of a 63-base linker (21 amino acids, with the sequence RVPRDPSGGGGSG GGGSGGGG) between EGFR and Halo.
T24 cells were transfected with the cDNA encoding EGFR-Halo, using a Nucleofector 2b device, and then starved in serumfree Ham's F12 medium for ≈16 h. The covalent complex between TMR and EGFR-Halo was formed in the same way as in the labeling of Halo-TfR. For EGF stimulation, HT medium containing 20 nM human EGF (Sigma-Aldrich) was added to the cells at a final concentration of 10 nM (1 ml of 20 nM EGF in HT medium was added to the glass-bottom dish containing 1 ml HT medium). The measurements were performed before and during 2.5-5 min after the EGF addition.
Ultrafast SFMI of fluorescent probes bound to coverslips and the molecules in the live-cell PM Each individual fluorescently labeled molecule was observed at 37 ± 1°C, using the oblique-angle and TIR illumination modes of a home-built objective lens-type TIRF microscope (based on an Olympus IX70 inverted microscope), which was modified and optimized for the camera system developed here. The beam was attenuated with neutral density filters, circularly polarized, and then steered into the edge of a high numerical aperture (NA) oil immersion objective lens (UAPON 150XOTIRF, NA = 1.45, Olympus), focused on the back focal plane of the objective lens. Both TIR and oblique-angle illuminations were used for Cy3 at 10 and 30 kHz, and 5xCy3-Tf at 45 kHz. For other molecules and conditions, only the TIR illumination was used. For exciting Cy3, Alexa555 (Invitrogen), JF549 (Janelia Fluor 549; Tocris Fujiwara et al. Journal of Cell Biology Bioscience), and TMR (Molecular Probes and Promega), a 532nm laser (Millennia Pro D2S-W, 2W, Spectra-Physics; 1.8-16 mW for the ≈14-µm diameter observation area) was employed. For exciting JF646 (Janelia Fluor 646; Tocris Bioscience), Atto647N (ATTO-TEC), SeTau647, and Cy5 (sulfo-Cy5; GE Healthcare), a 660-nm laser (Ventus, 750 mW, Laser Quantum; 0.41-4.5 mW for the ≈12-µm diameter observation area) was used. The illumination laser power density was determined as follows. When the field stop was fully open, the excitation laser illuminated a circular (2-dimensional Gaussian) area with a 12.5µm radius (standard deviation) on the sample plane. By adjusting the field-stop size, the central part of the excitation laser beam was selected to form a circular area with a radius of 7 µm on the sample plane, approximating the largest view-field of 14 × 14 µm 2 employed in this research, and the laser power after the objective lens was measured. Since the dimmest intensity in the 7-µm Gaussian area was ≈85% of the peak intensity, the laser power density was calculated assuming a uniform laser intensity in the circular area with a 7-µm radius; i.e., the measured power was simply divided by the area size of 154 µm 2 (π × 7 2 ).
To observe single fluorescent molecules immobilized on coverslips, dye molecules were covalently linked to 3aminopropyltriethoxysilane (APS)-coated coverslips (customordered from Matsunami). Briefly, 0.2-1 nM succinimidyl ester-modified dye molecules in HT medium were placed on the 3-APS-coated coverslip at 25°C (10-min incubation), and then the coverslip was washed five times with 1 ml HT medium. To immobilize 5xCy3-Tf molecules on the glass surface, the 12-mm diameter glass-bottom dishes were coated with poly-D-lysine, and then incubated in HT medium containing 0.1 nM 5xCy3-Tf at 25°C for 10 min.
The method for evaluating the position determination precisions of Cy3-labeled molecules bound on the coverslips is described in the caption to Fig. S2. The position determination precisions for the molecules in the apical PM were estimated using the ensemble-averaged MSD-Δt plots for single-molecule trajectories of Cy3-DOPE and Cy3-Tf (5xCy3-Tf) bound to TfR, as shown in Fig. S4.
To observe fluorescent molecules bound to the phospholipid and transmembrane proteins TfR and EGFR in the apical and basal PMs of live cells, the oblique-angle and TIR illumination modes were employed, respectively. To observe single molecules located outside the focal adhesion in the basal PM using ultrafast SFMI, the focal adhesion was visualized using mGFP-paxillin. Immediately before conducting ultrafast SFMI, the mGFPpaxillin image was obtained in the same view-field using the TIR illumination (0.063 µW/µm 2 at the specimen using a Spectra-Physics (Cyan-PC5W) 488-nm laser; using a frame rate of 60 Hz, averaged over 10 s).
Quantitative analysis of single-molecule trajectories based on the MSD-Δt plots (Fig. 4, A a and B) For each single-molecule trajectory, the one-dimensional MSD(x or y) for the x-or y-direction for every time interval was calculated according to the following formula: where δt is the frame time, (x(jδt + nδt) and y(jδt + nδt)) describe the position of the molecule following a time interval nδt after starting at position (x(jδt),y(jδt)), N is the total number of frames in the sequence, n and j are positive integers, and n determines the time increment. In the quantitative analysis of the trajectories, 1-dimensional MSD-Δt plots were employed. Each single-molecule trajectory was classified into a suppressed-, simple-Brownian-, or directed-diffusion mode using the 2-dimensional MSD-Δt plot, which is the sum of the MSD-Δt plots for the x-and y-directions (Fig. 4, A a). The basic idea for the classification is shown in Fig. 4, B a . The classification was based on the relative deviation (RD), which describes the long-term deviation of the actual mean-square displacement, MSD(N,n) at the time nδt, from the expected MSD based on the initial slope of the MSD-Δt plot for molecules undergoing ideal simple-Brownian diffusion, 4D 2−4 nδt; i.e., RD(N,n) = MSD(N,n)/[4D 2−4 nδt] (N = the number of frames in a full trajectory, n = the increment number of frames used for the analysis with the MSD-Δt plot, 1 ≤ n ≤ N, and δt = duration of each frame. Therefore, Δt = nδt, which is the x-axis of the MSD(N,n)-Δt plot; and D 2−4 is the short-time diffusion coefficient determined from the slope of the second, third, and fourth points in the MSD-Δt plot). The RD value is <<, ≈, or >> 1, when the molecules are undergoing suppressed, simple-Brownian, or directed diffusion, respectively.
In Fig. 4, Fig. 4, B b, respectively; they depend on both N and n). Each experimental single-molecule trajectory was classified into the suppressed (confined and hop)-diffusion mode if its RD(N,n) value (shaded bar) was smaller than RD min (and into the directeddiffusion mode if its RD(N,n) value was larger than RD MAX ).
Monte Carlo simulations for the hop diffusion in the intact PM and simple-Brownian diffusion in the actin-depleted blebbed PM (Fig. 5 B) Monte Carlo simulations for hop diffusion were performed as described previously . The hop-diffusion simulation parameters used are the following (p represents the probability that a hop movement to an adjacent compartment takes place when the diffusing molecule enters the boundary). 5xCy3-Tf-TfR at 45 kHz and TfR at 6 kHz: D micro = 4.5 µm 2 /s, L = 100 nm, p = 0.00045. Cy3-DOPE at 10 kHz: D micro = 9 µm 2 /s, L = 100 nm, p = 0.00065.
The simulation parameters used for simple-Brownian diffusion in the actin-depleted blebbed PM are the following.
TfR at 6 kHz: D micro = 3.3 µm 2 /s. Cy3-DOPE at 10 kHz: D micro = 6.0 µm 2 /s. A Gaussian localization error of 50 nm was added (see Fig.  S4 B). The MSD-Δt plot ensemble-averaged over 1,000 trajectories was obtained, and the offset due to the localization error was estimated and subtracted based on the y-intercept (by extrapolation) of the linear-fit function for the second, third, and fourth steps in the MSD-Δt plot, as performed for experimental trajectories.
Online supplemental material Fig. S1 illustrates the method we used to determine the electron amplification gain of the image intensifier. Fig. S2 displays the relationship of the single-molecule localization precision with the number of detected photons/molecule/frame for various fluorescent dye molecules. Fig. S3 shows that the numbers of detected photons/molecule/frame from single molecules are saturated under stronger excitation laser intensities, and thus the improvement of single-molecule localization precision with an increase of the laser intensity is limited. Fig. S4 presents the MSD-Δt plots ensemble averaged over all trajectories obtained for Cy3 molecules immobilized on coverslips or diffusing in the apical and basal PM of T24 cells, providing estimates of singlemolecule localization errors. Fig. S5 shows the TILD method. Video 1 exhibits single Cy3 molecules covalently linked to the glass surface observed at 10 kHz. Video 2 shows single Cy3-DOPE molecules diffusing in the intact apical PM in a viewfield of 256 × 256 pixels, observed at 10 kHz. Video 3 shows enlarged views of single Cy3-DOPE molecules diffusing in the intact apical PM and the actin-depleted blebbed PM, observed at 10 kHz. Video 4 shows enlarged views of single TfR molecules, bound by Cy3-Tf, diffusing in the intact apical PM and the actindepleted blebbed PM, observed at 6 kHz. Video 5 shows single TfR (Halo-TMR) molecules diffusing in the basal PM outside (top movies) and inside (bottom movies) the focal adhesion regions marked by mGFP-paxillin, observed at 6 kHz. Video 6 shows single EGFR (Halo-TMR) molecules diffusing in the basal PM before and 2.5 min after the addition of 10 nM EGF, observed at 6 kHz. Supplemental theory 1 describes the expected distribution of the residency times based on the hop-diffusion theory we developed. Supplemental theory 2 describes the function describing the MSD-Δt plot for particles undergoing hop diffusion used for "hop-diffusion fitting."

Data availability
Data supporting the findings of this study are available from the corresponding author upon reasonable request. The codes for the TILD analysis and for calculating the MSD based on the theoretical equation describing hop diffusion (to be used for hop-diffusion fitting using the MATLAB nonlinear fit function) are freely available for academic use at https://github.com/ kusumi-unit/WinTILD/ and https://github.com/kusumi-unit/ hop-diffusion-function/, respectively. Our method for obtaining the coordinates of the observed single molecules is basically the same as that described by Crocker and Grier (1996). The MATLAB codes based on this method written by Blair and Dufresne in 2009 are available at https://site.physics.georgetown. edu/matlab/code.html. Our codes for this have been integrated into a much larger, complex software package and they cannot be extracted in a useful way. However, the entire software is available from the corresponding author upon reasonable request (we will provide personal guidance on how to use it, as about half of its manual and comments are written in Japanese). The camera system developed here can now be custom ordered from Photron. The camera systems are also made available in our laboratories upon reasonable request. Y. Nagai, K. Iwasawa, and A. Kusumi developed the ultrahighspeed camera system, T.K. Fujiwara, K. Iwasawa, T.A. Tsunoyama, and A. Kusumi developed an ultrafast SFMI station based on the newly developed camera system, and T.K. Fujiwara, T. Kalkbrenner, T.A. Tsunoyama, and A. Kusumi tested the camera system on the developed station. T.K. Fujiwara, T.A. Tsunoyama, K.G.N. Suzuki, and A. Kusumi Huang et al., 2013) for 1,000 frames obtained at 10 kHz. (B) Typical images of single photons obtained as a function of the intensifier gain, using the newly developed camera system. They were obtained by amplifying single electrons emitted from the image-intensifier photocathode by the arrivals of single photons (uniform Köhler illumination by the strongly attenuated halogen lamp of the microscope). The amplifications were 506, 8,100, and 129,600 (increases by a factor of 16 from left to middle and middle to right images). Note that the term "overall electron amplification (of the camera system)" always excludes the 40% quantum efficiency of the image intensifier photocathode throughout this report, because we discuss the amplification of the number of electrons from the photoelectrons emitted by the photocathode. The range of the gray levels of the images shown here (both B and C) was set for 8 bits from 0 to 550 electrons/pixel at the CMOS sensor (from black to white). (C) As an example, we show that an overall electron amplification of 506× by the camera system would not be sufficient for the consistent detection of single photons, due to the readout noise. (C a) A schematic figure showing a 7 × 7 pixelated image for a single detected photon amplified to a total of 517 electrons (without noise and background; computer-generated, assuming an overall amplification of 506× with an SD of 0.28 ± 0.085 pixels as in B left). The number of amplified electrons in each pixel on the CMOS sensor is shown. A full well of 45,000 electrons/pixel of the CMOS sensor (SA1) is scaled to 12 bits (4,095 camera counts, and hence a unit camera count = 10.99 [≈11] electrons per pixel), and thus each number is a multiple of 11 electrons. (C b) Three arbitrarily selected (experimentally obtained) 7 × 7 pixelated images representing the spatial distributions of the readout noise of the CMOS sensor employed in this study (37-root-mean-square electrons/pixel/frame; see "Ultrahigh-speed intensified CMOS camera system: Design and operation" in Materials and methods). To detect a (photon-converted) emitted electron, its image, such as that shown in a, must be detectable in the presence of spatiotemporally varying noise, as shown here. The detectability will be enhanced by an increase of the electron amplification. See D. (D) Stochastic gain variations (fluctuations) of the image intensifier. At the level of detecting single photons, the gain variations are large (at the level of detecting single molecules, relative variations will become smaller due to averaging over all detected photons). In these histograms, the distributions of the number of total electrons stored at the CMOS sensor of the camera system for each single detected photon at the image-intensifier photocathode (i.e., for each discernible spot in images like those in B) are shown for the overall electron amplifications of 506×, 8,100×, and 129,600× (the y axis is normalized by the peak value in each histogram; the full x scale is increased by a factor of 16 from the left figure to the middle figure and from the middle figure to the right figure). The spots in the images (like those in B) were identified and the total number of electrons in each spot was evaluated using the functions of the ThunderSTORM plugin of ImageJ (Ovesný et al., 2014). For the spot detection with localization precisions at the pixel level (the peak pixel), we employed the "Wavelet filtering" (B-Spline order = 3 and B-Spline scale = 2.0) and the "Local maximum method" (Peak intensity threshold = 4.5 and Connectivity = 8-neighborhood). For the determination of the total number of electrons in a spot (and subpixel localization of each spot), we performed the Gaussian fitting of the image (i.e., the number of electrons/pixel in 11 × 11 pixels surrounding the peak pixel) and then integrated the best-fit function, using the Subpixel localization of molecules. These histograms were obtained using six 5,000-frame image sequences recorded at 10 kHz with a frame size of 13.4 × 13.4 µm on the focal plane, detecting 62,066, 459,156, and 510,165 photons (electrons emitted from the photocathode of the image intensifier) for overall amplifications of 506×, 8,100×, and 129,600×, respectively. They represent both the stochastic gain variations and the detectability of a photon image produced by the amplified electrons. See the histogram for 506×. The occurrences of the number of amplified electrons/detected photon sharply decreased when the number of amplified electrons was reduced below ≈650 electrons (the peak in the histogram). This is very likely due to a sharp reduction in the detectability of the spots produced by <650 electrons. Namely, when the signal intensity (the number of electrons) after amplification is small, the chance that the signal becomes less than the readout noise increases, because the readout noise pattern also fluctuates spatiotemporally (see C). (E) The probability of detecting a single photon was increased to 90.0% of the saturation level, at an overall electron amplification of 8,100×. Here, the number of detected photons per frame (mean ± SD); i.e., the number of discernible spots in the images like those in B (detected by the method described in D), is plotted as a function of the overall electron amplification (averaged over six 5,000-frame image sequences). With an increase in the overall electron amplification, the number of discernible spots increases. At an overall electron amplification of 8,100×, the number of discernible spots was 90.0% of the saturated number of spots (i.e., at the maximal overall electron amplification possible with the present instrument, which is 129,600× amplification). At the overall amplifications giving the saturated number of spots, virtually every photoelectron emitted from the photocathode is considered to be detected. Therefore, throughout the present research, we employed 8,100× as the overall electron amplification of the image intensifier (see "Ultrahigh-speed intensified CMOS camera system: Design and operation" in Materials and methods). Figure S2. Establishing the photophysics of various fluorescent probe molecules by independently evaluating the number of detected photons (proportional to the emitted photons) from a single fluorescent molecule during a single frame time (N; x-axis) and single-molecule localization precision (σ xy [σx + σ y]/ 2; y-axis), under various excitation laser powers and single-frame durations. These plots can be fitted well with the equation later in this legend, indicating that these measurements were performed with satisfactory accuracies. With an increase of the excitation laser power (at the sample), some dyes emit more photons than others, showing that they are more suitable for ultrafast SFMI. These curves are useful for determining the fluorescent probes to be used in the experiments and for predicting the single-molecule localization precisions that can be obtained under the given excitation laser powers. The results for 30, 10, and 0.06 kHz; i.e., the frame times of 0.033, 0.1 and 16.7 ms, respectively, are shown (45 kHz/0.022 ms for 5xCy3-Tf is also shown). For the method to evaluate N, see the subsection "Determination of the number of detected photons/molecule/frame (N)" in Materials and methods. In high-speed single fluorescent-molecule imaging, one of the crucial problems is whether single fluorescent molecules emit sufficient numbers of photons (during a single frame time) required for obtaining the desired single-molecule localization precisions. The results shown here demonstrate that a 10-kHz frame rate is applicable for various dye molecules, and Cy3 could even be used at 30 kHz. These plots were fitted well by the theoretical equation derived previously (Mortensen et al., 2010), indicating that the developed camera system functions as planned, even at high frequencies. The excess noise factor (F) of the developed camera system was evaluated by this fitting. Throughout this report (except for the measurements in the plasma membrane (PM), as described in Fig. S4), the localization precision (σ xy ) is defined as [σ x + σ y ]/2, where σ x and σ y are the standard deviations of the x and y position determinations, respectively, following the convention of the super-resolution imaging field (Dietrich et al., 2002;Martin et al., 2002). σ xy was determined in 15 consecutive frames for n = 50 trajectories for each condition. All of the fluorescent dye molecules were covalently bound to coverslips coated with 3-aminopropylethoxysilane, and 5xCy3-Tf was adsorbed on the coverslip coated with poly-D-lysine (Materials and methods). (A and B) Plots for single Cy3 molecules observed at 10 and 30 kHz and single 5xCy3-Tf molecules observed at 45 kHz (A) and those for various fluorescent molecules observed at 10 kHz (B). Five TIR laser illumination intensities were employed for each dye (50 molecules for each laser intensity), as indicated by the different colors of the data points. Various ranges of the laser power densities were used for different dyes, because the dyes are saturated differently (shown in each box). The plots (σ xy vs. N) shown in A and B could be fitted well (non-linear least-squares fitting by the Levenberg-Marquardt algorithm) using the following equation derived previously (Mortensen et al., 2010).
, where F is the sole fitting parameter, representing the excess noise factor (a coefficient describing the stochastic gain fluctuation in the electron amplification process in the image intensifier; F is shown in each box, but its value is 1.2-1.4 for all cases), s is the standard deviation of the Gaussian spot profile, 123 ± 1.1 nm for Cy3 on the sample plane (determined by the Gaussian fitting of each image for 50 Cy3 molecules immobilized on the glass excited by the TIR illumination at 79 µW/µm 2 ; compared with our standard condition of the oblique illumination at 23 µW/µm 2 , these observation conditions provided ∼3 times more detected photons; see Fig. 1 E, top; note that s depends on the observed fluorescent molecules), a is the pixel size (55.1 nm), and b is the standard deviation of the background noise. (For example, 0.038 ± 0.059 detected photons/pixel/frame [mean ± SD] for the TIR illumination and 0.035 ± 0.058 detected photons/pixel/frame for the oblique illumination at 10 kHz; n = 76,800 pixels = 32 × 32 pixels × 15 frames x 5 different positions.) The estimated excess noise factor F of the image intensifier shows that it is comparable to or slightly smaller (less noisy) than that of the EM-CCD electron multiplier (F = 1.4). Cy3 exhibited the least tendency to saturate, and thus provided better single-molecule localization precisions, consistent with the analysis results shown in Fig. S3, A and C. Note that s and b were determined for each fluorescent probe (with different illumination and excitation wavelengths and optics). 5xCy3-Tf data are considered to represent fluorescent spots generated by various numbers of Cy3 molecules placed within a few nanometers, mostly in the range of 3 to 8 molecules (1 and 2 Cy3 molecules/Tf, representing ≈12% of the 5xCy3-Tf spots, gave low signals, inducing extremely large errors in single-molecule localizations; meanwhile, the probability of 9 or more Cy3 molecules being attached to a Tf molecule will be <7%). Due to the photobleaching of multiple Cy3 molecules bound to a Tf molecule, the numbers detected on a Tf molecule decreased quickly upon laser illumination. (C) Plot for single Cy3 molecules observed at 60 Hz, with TIR illumination laser power densities ≤0.16 µW/µm 2 (indicated by different colors of the data points; 50 molecules for each laser intensity). The excess noise factor F was estimated to be 1.2, consistent with the results shown in A.
(D) Summary plot for single Cy3 molecules observed at 60 Hz, with TIR illumination laser power densities up to 79 µW/µm 2 : 0.018, 0.029, 0.047, 0.088, 0.16 (employed for the plot in c), 0.48, 1.6, 4.8, 14, 23, 43, and 79 µW/µm 2 (note that in this plot, in contrast to the others, the x-axis is in the log scale). The single-molecule localization precisions obtained with the laser power densities equal to and >14 µW/µm 2 were calculated using the equation above with F = 1.2, as found in C. This method for obtaining the single-molecule localization precisions employed here is different from that used for evaluating the precisions shown in Fig. 1 F and Fig. S2, A-C; and Fig. S3, B and C. Since most Cy3 molecules were photobleached within a single 16.7 ms frame, the more-prevalent method could not be employed. The x-axis of this figure covers the entire practical scale for the number of detected photons/ molecule/frame (N) for a single Cy3 molecule, from 25.0 ± 1.4 at a laser power density of 0.018 µW/µm 2 up to 11,400 ± 700 at 79 µW/µm 2 (mean ± SEM). This upper limit was given by the photobleaching and excitation power saturation of Cy3, and provided the best single-molecule localization precision of 2.6 ± 0.099 nm (mean ± SEM) for Cy3 (no further improvements could be obtain even by employing higher laser intensities; Fig. 1, G and H). Figure S4. MSD-Δt plots ensemble averaged over all trajectories, obtained by ultrahigh-speed single-molecule imaging of Cy3 molecules immobilized on coverslips or diffusing in the apical and basal PMs of T24 cells (using oblique-angle and TIR illuminations, respectively), providing estimates of single-molecule localization errors for diffusing molecules (as well as immobile molecules). All SEMs, including the error bars, are shown in the figure. The purposes of showing these figures are (1) to explain how to determine the single-molecule localization precisions of diffusing molecules in the PM using the MSD-Δt plot (because the method described in the caption to Fig. S2 is only useful for immobilized molecules) and (2) to show the actual localization precisions of diffusing molecules in the apical and basal PMs. First, see the panels in the left column, showing the MSD-Δt plots for molecules immobilized on the glass. Experimental MSD-Δt plots even for immobile molecules are expected to exhibit an offset, due to the position determination error; i.e., the flat MSD-Δt plot with a constant value (against Δt), which equals 4σ xy 2 (where σ xy = [σ x + σ y ]/2; Dietrich et al., 2002;Martin et al., 2002). The linear fitting indeed showed that the slopes were ≈0; the localization precisions determined here for Cy3 on the glass at 10 and 30 kHz were 22 and 37 nm, respectively (TIR illumination at 79 µW/µm 2 ). These results are consistent with those for immobilized molecules determined by the first method described in  Table 1). Next, see the panels in the middle and right columns, showing the MSD-Δt plots for molecules undergoing diffusion in the PM. As shown in the panels in the left column (immobilized molecules), the MSD values almost reach a plateau (which is the offset value) by the second step (Δt = 0.2 and 0.066 ms for Cy3 at 10 and 30 kHz, respectively). This means that the offset value of the MSD-Δt plot for diffusing molecules in the PM can be estimated as the y-intercept (by extrapolation) of the linear-fit function for the second, third, and fourth Figure S5. The TILD method for detecting the moment (instance) when a diffusing molecule in the PM undergoes the hop movement from a compartment to an adjacent one in the PM. We developed an improved method for detecting the hop moment (instance). This method detects the Transient Increase of the effective Local Diffusion (TILD) in a single-molecule trajectory. TILDs are likely to occur when a molecule hops between two membrane unknown even for an idealized hop-diffusion model, in which molecules undergo free diffusion (in viscous media) in the presence of equally spaced, equi-potential semi-permeable diffusion barriers. Here, we demonstrate that the residency time distribution is given by the sum of exponential distributions, which can be approximated well by a single exponential function with a decay constant of L 2 /4D MACRO , if the confinement effect is strong. Consider a Brownian particle in a square region with semi-permeable boundaries, and calculate the distribution of first exit times. The residency time distribution is analogous to the distribution of the first time a particle exits a certain region. We assume that diffusion along the x and y directions is independent, which allows us to solve the problem for a one-dimensional (1D) system, and then generalize the results to the two-dimensional (2D) case.
Consider a Brownian particle in 1D, initially placed at x = x 0 between two boundaries located at x = 0 and x = L, that can diffuse freely in this bounded region. When the particle attempts to leave this region, it faces resistance, and thus the boundaries can be considered as partially permeable barriers. Once the particle crosses the boundary, it can never go back. Therefore, the time when it leaves the region is always the first exit time. The probability distribution of such a particle is governed by the diffusion equation, with the boundary conditions, where D is the microscopic diffusion coefficient within a domain, and p is a constant related to the permeability of the boundaries, such that p→0 and p→∞ correspond to impenetrable and completely permeable boundaries, respectively. The solution can easily be obtained using separation of variables (Carslaw and Jaeger, 1986), and is given in the following form: ρ(x, t) X ∞ n 1 β n (x0)e −λ 2 n Dt L 2 ϕ n (x), ϕ n (x) γ n (αλncos(λnx L) + sin(λ n x L)), If the particle is initially at x = x 0 such that ρ(x,t) = δ(x−x 0 ), then the coefficient β n becomes β n ϕ n (x0) (17) so that ρ(x, t) X ∞ n 1 e −λ 2 n Dt L 2 ϕ n (x0)ϕ n (x). (18) The cumulative probability distribution can be expressed as Finally, the distribution of exit times f ex (t), and its cumulative F ex (t), can be calculated from the cumulative probability distribution above (Redner, 2001): Now we need to generalize this result to two dimensions and obtain the distribution of exit times from a square compartment of area L 2 . This was readily achieved by realizing that the time of the first exit is the minimum of the first exit time in the x and y directions. Using the well-known result for the distribution of the minimum of two independent random variables (Gumbel, 2004), we obtain the cumulative probability distribution for the exit time distribution as F ex( t) F ex,x( t) + F ex,y( t) − F ex,x( t)F ex,y( t), (21) where F ex,w (t) stands for the probability that the particle exits through one of the boundaries along the w direction until time t, and is explicitly given by where w 0 is the initial position along the w axis. Usually, the initial conditions are not accessible, so it is reasonable to average all of the different initial states. By averaging F ex,w (t) over all initial positions w 0 between 0 and L with equal weight, we obtain Using Eq. 23, Eq. 21 further simplifies to F ex( t) F ex,w( t)(2 − F ex,w( t)), as there is no difference in the statistics of the position between the x and y directions after averaging over the initial position. The probability distribution for the exit times, or the residency time distribution, is given by the first derivative of Eq. 24 with respect to time: where η n λ 2 n D/L 2 . The contributions of the higher order terms in Eq. 25 are quickly minimized, due to the exponential term. In many cases of practical interest, the first term with n = 1,m = 1 alone could be a good approximation of the result. To assess the validity of this argument, let us consider the ratio of the exponential factors in the first two terms of the double summation in Eq. 25. This ratio is equal to e − ( λ 2 2 −λ 2 1 ) Dt L 2 . (26) Inspecting the equation for λ n s, Eq. 16, we notice that λ n+1 ≈λ n +π, such that Therefore, the higher order terms decay with almost the 10th or greater powers of e −Dt/L 2 , which are always <1 for t > 0. Similarly, it can be shown that the coefficient of the exponential, ξ n ξ m η m , also decreases as n and m increase.
When the confinement effect is strong, the Brownian particle spends a sufficient amount of time in each compartment to cover it uniformly before escaping. Mathematically, this corresponds to the limit α 1, (28) in which ξ n is approximately equal to 1. Thus, the residency time distribution given in Eq. 25 can be approximated by a single exponential τ L 2 2Dλ 2 1 .
As the confinement effect becomes stronger, we would expect the average residency time to diverge. Therefore, in the strong confinement limit, λ 1 would be close to 0 such that tanλ 1 ≈λ 1 , and Eq. 16 can be replaced by an approximate form We now need to express α in terms of a quantity that can be experimentally measured. In the strong confinement limit that we are interested in, the Brownian particle behaves much like a random walker in a 2D lattice with lattice spacing L, for durations longer than the average residency time. In this picture, each lattice site corresponds to a square compartment of area L 2 , and the random walker takes steps between adjacent lattice sites at a rate F 1 4 τ. As the random walker can move in four directions in 2D, the rate of escape is 4F, such that the average residency time is τ. This description is appropriate as long as the random walker explores most of each compartment before it leaves. This ensures that the probability distribution within each compartment quickly becomes uniform and that the escape time distribution can be well approximated by a single exponential. The probability of finding the random walker at a lattice site (m,n) is governed by a Master equation (Hughes, 1995): dP m,n( t) dt FP m+1,n( t) + FP m−1,n( t) + FP m,n+1( t) + FP m,n−1( t) − 4FP m,n( t). (34) With a straightforward calculation, the mean square displacement of a random walker that is initially at the origin is given by r 2 X m,n L 2 P m,n t ( ) m 2 + n 2 ( ) 4 L 2 4τ t. (35) As this description is appropriate for times longer than τ, the diffusion coefficient associated with this random walk is the macroscopic diffusion coefficient, This value satisfies the condition for strong confinement, consistent with Eq. 28. Accordingly, under the strong confinement conditions, the second order term proportional to α −2 in Eq. 33 can be neglected, and Eq. 33 can simply be expressed as Therefore, the decay time constant of the exponential distribution of the residency time within a compartment, given by Eq. 33, can be simplified to the expression given by Eq. 40.
We obtained the distribution of the dwell lifetimes in a compartment by directly measuring the dwell time using the TILD analysis (Fig. S5, A c and B c). In the real PM, the dwell time of diffusing molecules within a compartment would be affected by the variations in the compartment sizes and shapes and in the properties of the compartment boundaries, due to the differences in actin binding proteins and TM picket proteins. However, as shown in Fig. S5 B c and Fig. 5 A, the residency time distribution in the PM is well approximated by an exponential function, probably because these variations both lengthen and shorten the dwell lifetimes quite randomly.
Supplemental theory 2. Hop-diffusion fitting: The function describing the MSD-Δt plot for particles undergoing hop diffusion We developed an equation for the MSD-Δt plot for particles undergoing idealized hop diffusion; i.e., free diffusion in the presence of equally spaced, equi-potential semi-permeable diffusion barriers. Such hop diffusion can be characterized by the following three parameters: the compartment size (the distance between barriers), L, the microscopic diffusion coefficient within a compartment (true diffusion coefficient in the absence of the compartments), D micro , and the long-term diffusion coefficient over many compartments, D MACRO . One of the key parameters for hop diffusion, the residency time within a compartment, can be calculated from these parameters as L 2 /4D MACRO , as shown in Supplemental theory 1.
By employing the notations γ = D MACRO /(D micro −D MACRO ) and τ = 4D micro t/L 2 , the 1-dimensional MSD in the real time domain, which is the inverse Laplace transform of Eq. 18 in Kenkre et al. (2008) for a 1-dimensional MSD averaged over all initial locations, can be obtained as where the terms res 0 (τ) and res(τ) are the sum of the residues that arise from the inverse Laplace transform. The zeroth residue, res 0 (τ), is expressed as res 0( τ) 3τγ + γ + 3τ 3(γ + 1) 2 . (42) whereas the term res(τ), representing half the sum of all other residues, is expressed as res(τ) − X N k 2 e −τr 2 (k) (γtan 2 (r(k)) + γ + 1)r 2 (k) , where r(k) (k = 2,3,4,...) is the k th root of − 1 γ y tan(y), for y ≥ 0. As performed in Kenkre et al. (2008), we employed the bisection method to find the roots numerically with high precisions and obtained the accurate result for a sufficiently large N. Namely, by rearranging terms in Eq. 43, we obtain the following equation.
We used this equation for fitting the experimental MSD values. As for the physical meaning, the first term characterizes the long-term behavior of MSD, which is approximately equal to 2D MACRO t for strong confinement (γ ≈ 0) and 2D micro t for weak confinement (γ →∞). The second and third terms describe the transient behavior of the MSD. At around τ ≈ 0, the second and third terms cancel each other out. In the longer time limit, the third term becomes negligible, and the second term becomes equal to the intercept of the linear MSD.